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

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Johnson Polyhedra