  The critical group of a finite graph is an abelian group defined by the Smith
normal form of the Laplacian. We determine the the critical groups of the
Peisert graphs, a certain family of strongly regular graphs similar to, but
different from, the Paley graphs. It is further shown thatthe adjacency
matrices of the two graphs defined over a field of order $p^2$ with $p\equiv
3\pmod 4$ are similar over the $\ell$-local integers for every prime $\ell$.
Consequently, each such pair of graphs provides an example where all the
corresponding generalized adjacency matrices are both cospectral and equivalent
in the sense of Smith normal form.
