What is the fundamental group of the octahedral symmetry group Oh? Do discrete groups have a well defined topology? If so, what is the fundamental group of the octahedral symmetry group Oh? In other words, are all the maps of one-spheres on the symmetry elements of Oh continuously deformable into each other?

Of course it has a well-defined topology...the discrete one (all subsets are open subsets).

The fundamental group requires a choice of base point. Given that, of course all the base-point-preserving maps are continuously deformable to each other, since there's only one of them! So the fundamental group is trivial.