Recognizing the alternating groups from their probabilistic zeta function

Let G be a finite group; there exists a uniquely determined Dirichlet polynomial P(G,s) such that if t is an element of N, then P(G,t) gives the probability of generating G with t randomly chosen elements. We show that if P(G,s) = P(Alt(n),s), then G/Frat G congruent to Alt(n)