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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[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},
   {Sqrt[(5 + 2*Sqrt[5])/5], 0, (-5 - Sqrt[10*(5 + Sqrt[5])])/10}, {Sqrt[(5 + 2*Sqrt[5])/5], 0,
    (5 + Sqrt[10*(5 + Sqrt[5])])/10}, {Sqrt[1/8 + 1/(8*Sqrt[5])], (-3 - Sqrt[5])/4,
    (-5 - Sqrt[10*(5 + Sqrt[5])])/10}, {Sqrt[1/8 + 1/(8*Sqrt[5])], (-3 - Sqrt[5])/4,
    (5 + Sqrt[10*(5 + Sqrt[5])])/10}, {Sqrt[1/8 + 1/(8*Sqrt[5])], (3 + Sqrt[5])/4,
    (-5 - Sqrt[10*(5 + Sqrt[5])])/10}, {Sqrt[1/8 + 1/(8*Sqrt[5])], (3 + Sqrt[5])/4,
    (5 + Sqrt[10*(5 + Sqrt[5])])/10}, {-Sqrt[5/8 + 11/(8*Sqrt[5])], (-1 - Sqrt[5])/4,
    (-5 - Sqrt[10*(5 + Sqrt[5])])/10}, {-Sqrt[5/8 + 11/(8*Sqrt[5])], (-1 - Sqrt[5])/4,
    (5 + Sqrt[10*(5 + Sqrt[5])])/10}, {-Sqrt[5/8 + 11/(8*Sqrt[5])], (1 + Sqrt[5])/4,
    (-5 - Sqrt[10*(5 + Sqrt[5])])/10}, {-Sqrt[5/8 + 11/(8*Sqrt[5])], (1 + Sqrt[5])/4,
    (5 + Sqrt[10*(5 + Sqrt[5])])/10}, {-Sqrt[1/2 + 1/(2*Sqrt[5])], 0, (-5 - 2*Sqrt[5*(5 + 2*Sqrt[5])])/10},
   {-Sqrt[1/2 + 1/(2*Sqrt[5])], 0, (5 + 2*Sqrt[5*(5 + 2*Sqrt[5])])/10},
   {-Sqrt[(5 - Sqrt[5])/10]/2, (-1 - Sqrt[5])/4, (-5 - 2*Sqrt[5*(5 + 2*Sqrt[5])])/10},
   {-Sqrt[(5 - Sqrt[5])/10]/2, (-1 - Sqrt[5])/4, (5 + 2*Sqrt[5*(5 + 2*Sqrt[5])])/10},
   {-Sqrt[(5 - Sqrt[5])/10]/2, (1 + Sqrt[5])/4, (-5 - 2*Sqrt[5*(5 + 2*Sqrt[5])])/10},
   {-Sqrt[(5 - Sqrt[5])/10]/2, (1 + Sqrt[5])/4, (5 + 2*Sqrt[5*(5 + 2*Sqrt[5])])/10},
   {Sqrt[1/4 + 1/(2*Sqrt[5])], -1/2, (-5 - 2*Sqrt[5*(5 + 2*Sqrt[5])])/10},
   {Sqrt[1/4 + 1/(2*Sqrt[5])], -1/2, (5 + 2*Sqrt[5*(5 + 2*Sqrt[5])])/10},
   {Sqrt[1/4 + 1/(2*Sqrt[5])], 1/2, (-5 - 2*Sqrt[5*(5 + 2*Sqrt[5])])/10},
   {Sqrt[1/4 + 1/(2*Sqrt[5])], 1/2, (5 + 2*Sqrt[5*(5 + 2*Sqrt[5])])/10}},
  Polygon[{{36, 32, 34, 38, 40}, {30, 32, 36}, {28, 34, 32}, {24, 38, 34}, {22, 40, 38}, {26, 36, 40},
    {12, 4, 26}, {8, 16, 30}, {14, 6, 28}, {2, 10, 24}, {18, 20, 22}, {4, 8, 30, 36, 26},
    {16, 14, 28, 32, 30}, {6, 2, 24, 34, 28}, {10, 18, 22, 38, 24}, {20, 12, 26, 40, 22},
    {39, 37, 33, 31, 35}, {35, 31, 29}, {31, 33, 27}, {33, 37, 23}, {37, 39, 21}, {39, 35, 25}, {25, 3, 11},
    {29, 15, 7}, {27, 5, 13}, {23, 9, 1}, {21, 19, 17}, {25, 35, 29, 7, 3}, {29, 31, 27, 13, 15},
    {27, 33, 23, 1, 5}, {23, 37, 21, 17, 9}, {21, 39, 25, 11, 19}, {11, 3, 4, 12}, {3, 7, 8, 4},
    {7, 15, 16, 8}, {15, 13, 14, 16}, {13, 5, 6, 14}, {5, 1, 2, 6}, {1, 9, 10, 2}, {9, 17, 18, 10},
    {17, 19, 20, 18}, {19, 11, 12, 20}}]]]

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Johnson Polyhedra