Finite generation of iterated wreath products

Let $(G_n,X_n)$ be a sequence of finite transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated permutational wreath product $\mathrm{.}\mathrm{.}\mathrm{.}\wr G_2\wr G_1$ is topologically finitely generated if and only if the profinite abelian group ${\prod }_{n\ge 1}{G}_n\mathrm{/}{G}_n^{\mathrm{\prime }}$ is topologically finitely generated. As a corollary, for a finite transitive group $G$ the minimal number of generators of the wreath power $G\wr…\wr G\wr G$($n$times)isboundedif$G$isperfect,andgrowslinearlyif$G$ is non-perfect. As a by-product we construct a finitely generated branch group, which has maximal subgroups of infinite index, answering [2,Question 14].