What is the current known hardness of Graph Isomorphism? 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.
Graph Isomorphism is hard for DET, the class of problems which are $NC^1$ reducible to the determinant. See "On the Hardness of Graph Isomorphism" by Jacobo Toran. http://epubs.siam.org/sicomp/resource/1/smjcat/v33/i5/p1093_s1?isAuthorized=no

It seems this is the strongest hardness result to date.