Let G be a finite almost simple group. It is well known that G can be generated by three elements,
and in previous work we showed that 6 generators suffice for all maximal subgroups of G. In
this paper, we consider subgroups at the next level of the subgroup lattice—the so-called second
maximal subgroups. We prove that with the possible exception of some families of rank 1 groups
of Lie type, the number of generators of every second maximal subgroup of G is bounded by an
absolute constant. We also show that such a bound holds without any exceptions if and only if there
are only finitely many primes r for which there is a prime power q such that (q
r − 1)/(q − 1)
is prime. The latter statement is a formidable open problem in Number Theory. Applications to
random generation and polynomial growth are also given.