  We consider the critical group of a hypothetical Moore graph of diameter $2$
and valency $57$. Determining this group is equivalent to finding the Smith
normal form of the Laplacian matrix of such a graph. We show that all of the
Sylow $p$-subgroups of the critical group must be elementary abelian with the
exception of $p = 5$. We prove that the $5$-rank of the Laplacian matrix
determines the critical group up to two possibilities.
