Let $G$ be a finite $p$-group of nilpotency class $c$ and of derived length $d$. As is well known, we have $d\leq [\log_2 c]+1$,

(https://groupprops.subwiki.org/wiki/Derived_length_is_logarithmically_bounded_by_nilpotency_class).

Does there exist any information or any classification of finite $p$-group $G$, where $d=[\log_2 c]+1$?

Any answer or comment will be greatly appreciated!