A finiteness condition on the coefficients of the probabilistic zeta function

We discuss whether finiteness properties of a profinite group $G$ can be deduced from the coefficients of the probabilistic zeta function $P_G(s)$. In particular we prove that if $P_G(s)$ is rational and all but finitely many non abelian composition factors of $G$ are isomorphic to $PSL(2,p)$ for some prime $p$, then $G$ contains only finitely many maximal subgroups.