What is this thing?

I made my first truncated icosahedron (a.k.a "soccer ball") out of the magnetic balls.  I just built little rings of 5 and 6 balls, and stuck them together so that each vertex had two hexagons and one pentagon.  This resulted in little squares being formed where the faces met, and little triangles around what I considered the vertices (of the truncated icosahedron).  I wondered what this polyhedron would be called if each ball were a vertex.  A quick Google search reminded me that I know very little about this subject, and I wasn't able to find the answer.

Can you name this polyhedron?