A chain of normalizers in the sylow $$\mathbf{2}$$-subgroups of the symmetric group on $${\mathbf{2}}^{\varvec{n}}$$ letters

On the basis of an initial interest in symmetric cryptography, present work we study a chain subgroups. Starting from Sylow $2$-subgroup AGL(2,n), each term is defined as normalizer previous one group on $2^n$ letters. Partial results and computational experiments lead us to conjecture that, for large values $n$, index consecutive does not depend $n$. Indeed, there strong evidence that sequence logarithms such indices partial sums numbers partitions into at least two distinct parts.