Closures of Finite Primitive Permutation Groups

Let G be a primitive permutation group on a finite set Ω, and, for k ⩾ 2, let G(k) be the k-closure of G, that is, the largest subgroup of Sym (Ω) preserving all the G-invariant k-relations on Ω. Suppose that G<H⩽G(k) and G and H have different socles. It is shown that k⩽5 and the groups G and H are classified explicitly.