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 ${\ldots\wr G_2\wr G_1}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 ?\wr G₂\wr G₁{\ldots\wr G_2\wr G_1} is topologically finitely generated if and only if the profinite abelian group ?_{n 3 1} G_n/G￠_n{\prod_{n\geq 1} G_n/G'_n} 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{G\wr \ldots\wr G\wr G} (n times) is bounded if G is perfect, and grows linearly if G is non-perfect. As a by-product we construct a finitely generated branch group, which has maximal subgroups of infinite index.