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Graphics3D[GraphicsComplex[
  {{0, (-1 - Sqrt[5])/2, -1/2 + (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {0, (-1 - Sqrt[5])/2, 1/2 + (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {0, (1 + Sqrt[5])/2, -1/2 + (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {0, (1 + Sqrt[5])/2, 1/2 + (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {Sqrt[1/8 + 1/(8*Sqrt[5])], (-3 - Sqrt[5])/4, (5 + Sqrt[10*(5 + Sqrt[5])])/10 +
     (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {Sqrt[1/8 + 1/(8*Sqrt[5])], (3 + Sqrt[5])/4, (5 + Sqrt[10*(5 + Sqrt[5])])/10 +
     (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {Sqrt[1/4 + 1/(2*Sqrt[5])], -1/2, (-5 - Sqrt[10*(5 - Sqrt[5])])/10 +
     (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {Sqrt[1/4 + 1/(2*Sqrt[5])], -1/2, (5 + 2*Sqrt[5*(5 + 2*Sqrt[5])])/10 +
     (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {Sqrt[1/4 + 1/(2*Sqrt[5])], 1/2, (-5 - Sqrt[10*(5 - Sqrt[5])])/10 +
     (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {Sqrt[1/4 + 1/(2*Sqrt[5])], 1/2, (5 + 2*Sqrt[5*(5 + 2*Sqrt[5])])/10 +
     (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {-Sqrt[1/2 + 1/(2*Sqrt[5])], 0, (-5 - Sqrt[10*(5 - Sqrt[5])])/10 +
     (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {-Sqrt[1/2 + 1/(2*Sqrt[5])], 0, (5 + 2*Sqrt[5*(5 + 2*Sqrt[5])])/10 +
     (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {-Sqrt[5/8 + 11/(8*Sqrt[5])], (-1 - Sqrt[5])/4, (5 + Sqrt[10*(5 + Sqrt[5])])/10 +
     (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {-Sqrt[5/8 + 11/(8*Sqrt[5])], (1 + Sqrt[5])/4, (5 + Sqrt[10*(5 + Sqrt[5])])/10 +
     (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {-Sqrt[(5 - Sqrt[5])/10]/2, (-1 - Sqrt[5])/4, (-5 - Sqrt[10*(5 - Sqrt[5])])/10 +
     (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {-Sqrt[(5 - Sqrt[5])/10]/2, (-1 - Sqrt[5])/4, (5 + 2*Sqrt[5*(5 + 2*Sqrt[5])])/10 +
     (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {-Sqrt[(5 - Sqrt[5])/10]/2, (1 + Sqrt[5])/4, (-5 - Sqrt[10*(5 - Sqrt[5])])/10 +
     (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {-Sqrt[(5 - Sqrt[5])/10]/2, (1 + Sqrt[5])/4, (5 + 2*Sqrt[5*(5 + 2*Sqrt[5])])/10 +
     (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {-Sqrt[5/8 + Sqrt[5]/8], (-3 - Sqrt[5])/4, -1/2 + (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/
      14320}, {-Sqrt[5/8 + Sqrt[5]/8], (-3 - Sqrt[5])/4,
    1/2 + (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {-Sqrt[5/8 + Sqrt[5]/8], (3 + Sqrt[5])/4, -1/2 + (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/
      14320}, {-Sqrt[5/8 + Sqrt[5]/8], (3 + Sqrt[5])/4,
    1/2 + (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {Sqrt[5/8 + Sqrt[5]/8], (-3 - Sqrt[5])/4, -1/2 + (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/
      14320}, {Sqrt[5/8 + Sqrt[5]/8], (-3 - Sqrt[5])/4,
    1/2 + (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {Sqrt[5/8 + Sqrt[5]/8], (3 + Sqrt[5])/4, -1/2 + (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/
      14320}, {Sqrt[5/8 + Sqrt[5]/8], (3 + Sqrt[5])/4,
    1/2 + (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {-Sqrt[5/4 + Sqrt[5]/2], -1/2, -1/2 + (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {-Sqrt[5/4 + Sqrt[5]/2], -1/2, 1/2 + (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {-Sqrt[5/4 + Sqrt[5]/2], 1/2, -1/2 + (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {-Sqrt[5/4 + Sqrt[5]/2], 1/2, 1/2 + (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {Sqrt[5/4 + Sqrt[5]/2], -1/2, -1/2 + (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {Sqrt[5/4 + Sqrt[5]/2], -1/2, 1/2 + (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {Sqrt[5/4 + Sqrt[5]/2], 1/2, -1/2 + (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {Sqrt[5/4 + Sqrt[5]/2], 1/2, 1/2 + (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320},
   {Sqrt[(5 + 2*Sqrt[5])/5], 0, (5 + Sqrt[10*(5 + Sqrt[5])])/10 +
     (1660 + 422*Sqrt[5] - 3*Sqrt[4354450 + 775210*Sqrt[5]])/14320}},
  Polygon[{{18, 12, 16, 8, 10}, {14, 12, 18}, {13, 16, 12}, {5, 8, 16}, {35, 10, 8}, {6, 18, 10}, {26, 4, 6},
    {22, 30, 14}, {28, 20, 13}, {2, 24, 5}, {32, 34, 35}, {4, 22, 14, 18, 6}, {30, 28, 13, 12, 14},
    {20, 2, 5, 16, 13}, {24, 32, 35, 8, 5}, {34, 26, 6, 10, 35}, {7, 15, 11, 17, 9}, {3, 25, 9, 17},
    {29, 21, 17, 11}, {19, 27, 11, 15}, {23, 1, 15, 7}, {33, 31, 7, 9}, {9, 25, 33}, {17, 21, 3},
    {11, 27, 29}, {15, 1, 19}, {7, 31, 23}, {25, 3, 4, 26}, {3, 21, 22, 4}, {21, 29, 30, 22},
    {29, 27, 28, 30}, {27, 19, 20, 28}, {19, 1, 2, 20}, {1, 23, 24, 2}, {23, 31, 32, 24}, {31, 33, 34, 32},
    {33, 25, 26, 34}}]]]

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Johnson Polyhedra