  This paper concerns finite groups of class (at most) two and of odd prime
exponent $p$. Such a group is called special if the center lies within its
derived group. Every group of class 2 and exponent $p$ can be uniquely
expressed as the direct product of an elementary abelian group and a special
group. This reduces the isomorphism problem to special groups. The special
groups having $G'$ cyclic are well known. Groups having $|G'|=p^2$ are known to
be generated by two abelian subgroups. As such, they can be described by a pair
of Scharlau Matrices which we will define. Using these, Vishnevetskii ([1],
[2]) classified the special groups which are not central products of groups of
smaller order. We call these Vishnevetskii indecomposable. All decomposable
groups are central products of two or more indecomposable groups. By Theorem 2
of [2], if $G$ is a special group with derived group of order $p^2$ then the
indecomposable central factors of $G$, together with their multiplicities, form
a set of invariants for $G$. However these invariants do not determine $G$,
since (as we will show below) two nonisomorphic groups can have the same
indecomposable central factors. The groups of exponent $p$ and order dividing
$p^8$ are already in the literature. In this paper we prove that there are six
isomorphism types of special groups of order $p^9$ having $|G'|=p^2$, and we
list them.
  We will also introduce a way of representing groups of class 2 having
exponent $p$ by using digraphs with flows on the edges. The digraph is compact
and gives, on sight, a great deal of structural information about the group and
completely defines the group. Some group invariants will also be described
which are easy to compute from the digraphs and which can readily be used to
distinguish the isomorphism types for small orders.
