tri = {
  {3,5,6}, {3,2,5}, {3,6,4}, {3,0,2}, {3,4,1},
  {3,1,0}, {5,0,6}, {5,2,4}, {5,4,7}, {5,7,0},
  {6,7,4}, {6,0,1}, {6,1,7}, {2,1,4}, {2,0,7}, {2,7,1}
};

pts = {
  { 0.64,  -0.20,  1.0000000000000000},
  {-1.09,   0.38,  0.0206663266698444},
  {-0.25,   0.51,  0.0048531277065193},
  { 0.78,   0.62,  0.0082275214556137},
  {-0.78,  -0.62,  0.0082275214556137},
  { 0.25,  -0.51,  0.0048531277065193},
  { 1.09,  -0.38,  0.0206663266698444},
  {-0.64,   0.20,  1.0000000000000000}
};

Graphics3D[
  GraphicsComplex[pts, Polygon /@ (tri + 1)],
  Boxed -> False
]