it is the same as the ordinary symmetry group. (rotations and reflections). if the question was what's this group, then it is a group of order 48, you can get a presentation of it by labelling the vertices. It is abstractly isomorphic to S_4 x C_2, symmetric group on 4 letters direct product cyclic group of order 2.
To see the iso with S_4xC_2 imagine drawing in the diagonals of the box, there are 4 of them. the symmetries permute these 4 objects, the C_2 part is because there are reflections too.
If I remember my solid state course correctly, the group for a cube is [tex]O_{h} [/tex]. This takes into account the inversion symmetry. I believe it is written as [tex]O_{h} = S_{4} x C_{2} [/tex].