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Given a finite covering of graphs $f : Y \to X$, it is not always the casethat $H_1(Y;\mathbb{C})$ is spanned by lifts of primitive elements of$\pi_1(X)$. In this paper, we study graphs for which this is not the case, andwe give here the simplest known nontrivial examples of covers with thisproperty, with covering degree as small as 128. Our first step is focusing ourattention 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 foridentifying deck groups for which there exist covers with the property. Wepresent an algorithm for determining if a finite $p$-group satisfies thiscriterion that uses only the character table of the group. Finally, we providea complete census of all finite $p$-groups of rank $\geq 3$ and order $< 1000$satisfying this criterion, all of which are new examples.

Graph coverings and (im)primitive homology: some new examples of exceptionally low degree