  Given a finite covering of graphs $f : Y \to X$, it is not always the case
that $H_1(Y;\mathbb{C})$ is spanned by lifts of primitive elements of
$\pi_1(X)$. In this paper, we study graphs for which this is not the case, and
we give here the simplest known nontrivial examples of covers with this
property, with covering degree as small as 128. Our first step is focusing our
attention on the special class of graph covers where the deck group is a finite
$p$-group. For such covers, there is a representation-theoretic criterion for
identifying deck groups for which there exist covers with the property. We
present an algorithm for determining if a finite $p$-group satisfies this
criterion that uses only the character table of the group. Finally, we provide
a complete census of all finite $p$-groups of rank $\geq 3$ and order $< 1000$
satisfying this criterion, all of which are new examples.
