$A$ is a generating set of permutation-group $G$ (not a p-group), i.e. $\langle A \rangle=G$. $|A| \leq \log_2(|G|)$.
$A$ has possible maximum number of elements to generate $G$.
It means that the next possible generating set of $G$, say $C$, has cardinality $> \log(|G|)$. There is no generating set between $A$ and $C$.
Is there another set $B$ with the same cardinality of $A$ and $\langle B \rangle =G$ ?
- Note that every subgroup of $ Sym(n)$ (symmetric group) can be generated by at most $n$ elements , this bound has been lowered, using the classification, to $[ n /2 ]$ if $n > 3$ (P. Neumann).