Comparing the order and the minimal number of generators of a finite irreducible linear group

We denote the smallest size of a generating set of a group G by d(G). We prove that d(G)log⁡|G|=O(n2log⁡q) for irreducible subgroups G of GL(n,q), and estimate the associated constants. The result is motivated by attempts to bound the complexity of computing the automorphism groups of various classes of finite groups.