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

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Johnson Polyhedra