  A Cayley graph Cay$(G;S)$ has the CI (Cayley Isomorphism) property if for
every isomorphic graph Cay$(G;T)$, there is a group automorphism $\alpha$ of
$G$ such that $S^\alpha=T$. The DCI (Directed Cayley Isomorphism) property is
defined analogously on digraphs. A group $G$ is a CI-group if every Cayley
graph on $G$ has the CI property, and is a DCI-group if every Cayley digraph on
$G$ has the DCI property. Since a graph is a special type of digraph, this
means that every DCI-group is a CI-group, and if a group is not a CI-group then
it is not a DCI-group, but there are well-known examples of groups that are
CI-groups but not DCI-groups.
  In 2009, Spiga showed that $\mathbb Z_3^8$ is not a DCI-group, by producing a
digraph that does not have the DCI property. He also showed that $\mathbb
Z_3^5$ is a DCI-group (and therefore also a CI-group). Until recently the
question of whether there are elementary abelian $3$-groups that are not
CI-groups remained open. In a recent preprint with Dave Witte Morris, we showed
that $\mathbb Z_3^{10}$ is not a CI-group. In this paper we show that with
slight modifications, the underlying undirected graph of order $3^8$ described
by Spiga is does not have the CI property, so $\mathbb Z_3^8$ is not a
CI-group.
