Upper bounds of homological invariants of {FI_G}-modules

Given a finite group G, let P_G(s) be the probability that s randomly chosen elements generate G, and let H be a finite group with \({P_{G}(s) = P_{H}(s)}\). We show that if the nonabelian composition factors of G and H are PSL(2, p) for some non-Mersenne prime \({p \geq 5}\), then G and H have the same non-Frattini chief factors.