Inspired by the question is factoring known to be P-hard, I wonder what the current similar state of knowledge is about the hardness of graph isomorphism. I am sure that it is currently not known if GI is in P, but:
what is the currently known largest class that GI is harder than?
(it was not answered at a similar sounding question)
To address some of the comments, I want to know the currently known maximal class(es) that GI, the problem is complete for. Known algorithms for GI are upper-bounded by superpolynomial functions, and it is a member of NP. But it is not known that GI is P-hard. I'd like to know any classes C for which it -is- known it is C-hard, and hopefully as inclusive as possible.