On base sizes for almost simple primitive groups

Let G⩽Sym(Ω) be a finite almost simple primitive permutation group with socle G₀. A subset of Ω is a base for G if its pointwise stabilizer is trivial; the base size of G, denoted b(G), is the minimal size of a base. We say that G is standard if G₀=A_n and Ω is an orbit of subsets or partitions of {1,…,n}, or if G₀ is a classical group and Ω is an orbit of subspaces (or pairs of subspaces) of the natural module for G₀. The base size of a standard group can be arbitrarily large, in general, whereas the situation for non-standard groups is rather more restricted. Indeed, we have b(G)⩽7 for every non-standard group G, with equality if and only if G is the Mathieu group M₂₄ in its natural action on 24 points. In this paper, we extend this result by classifying the non-standard groups with b(G)=6. The main tools include recent work on bases for actions of simple algebraic groups, together with probabilistic methods and improved fixed point ratio estimates for exceptional groups of Lie type.