Weak Cayley tables and generalized centralizer rings of finite groups

For a finite group G we study certain rings (sic)(G)((k)) called k-S-rings, one for each k >= 1, where S (1) (sic)(G)((1)) is the centraliser ring Z(CG) of G. These rings have the property that (sic)(G)((k+ 1)) determines (sic)(G)((k)) for all k >= 1. We study the relationship of (sic)(G)((2)) with the weak Cayley table of G. We show that (sic)(G)((2)) and the weak Cayley table together determine the sizes of the derived factors of G (noting that a result of Mattarei shows that (sic)(G)((1)) = Z(CG) does not). We also show that (sic)(G)((4)) determines G for any group G with finite conjugacy classes, thus giving an answer to a question of Brauer. We give a criteria for two groups to have the same 2-S-ring and a result guaranteeing that two groups have the same weak Cayley table. Using these results we find a pair of groups of order 512 that have the same weak Cayley table, are a Brauer pair, and have the same 2-S-ring.