The reciprocal character of the conjugation action

For a finite group $G$, we investigate the smallest positive integer $e(G)$ such that the map sending $g\in G$ to $e(G)|G:\mathrm{C}_G(g)|$ is a generalized character of $G$. It turns out that $e(G)$ is strongly influenced by local data, but behaves irregularly for non-abelian simple groups. We interpret $e(G)$ as an elementary divisor of a certain non-negative integral matrix related to the character table of $G$. Our methods applied to Brauer characters also answer a recent question of Navarro: The $p$-Brauer character table of $G$ determines $|G|_{p'}$.