Let $n \geq 1$ and let $H_n$ be a 2-Sylow subgroup of the symmetric group $\mathrm{Sym}(2^n)$. Let also consider the cycle $\gamma_n = (1, \ldots, 2^n)$ of order $2^n$.

If we assume moreover that $H_n$ contains the transposition $(1,2)$ then $\Sigma_n = \{{\gamma_n^{\pm1}\}} \cup H_n$ is a generating set of $\mathrm{Sym}(2^n)$, and we can consider the word length on $\mathrm{Sym}(2^n)$ associated to $\Sigma_n$, defined by $$ |g|_{\Sigma_n} = \min \{ k \geq 0 : \exists s_1, \ldots, s_k \in \Sigma_n; \sigma = s_1 \cdots s_k \},$$ for $g \in \mathrm{Sym}(2^n)$.

Now let $m_n$ be the maximum of the $|g|_{\Sigma_n}$ when $g$ ranges over $\mathrm{Sym}(2^n)$.

**Question 1**: is the sequence $(m_n)_n$ bounded ? (this may not depend on the choice of the Sylow subgroup $H_n$)

(EDIT: the answer to Question 1 is no according to Peter Mueller's answer below)

If the answer to Question 1 is no, then

**Question 2**: are there known bounds for $m_n$ ?

(EDIT: I am interested in an upper bound, i.e. an upper bound of the diameter of the Cayley graph $\mathrm{Cay}(\mathrm{Sym}(2^n), \Sigma_n)$ of the symmetric group $\mathrm{Sym}(2^n)$.)