  In this paper, we show that the group $\mathbb{Z}_p^5$ is a DCI-group for any
odd prime $p,$ that is, two Cayley digraphs Cay$(\mathbb{Z}_p^5,S)$ and
Cay$(\mathbb{Z}_p^5,T)$ are isomorphic if and only if $S=T^\varphi$ for some
automorphism $\varphi$ of the group $\mathbb{Z}_p^5$.
