Let $G$ be a finite $p$-group of nilpotency class $c$ and of derived length $d$.
As is well known, we have $d\leq \lfloor\log\_2 c\rfloor+1$, 


(<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 $d=\lfloor\log\_2 c\rfloor+1$?


Any answer or comment will be greatly appreciated!



The nilpotency class and the derived length of a $p$-group
My guess is "no", at least as regards a classification. A check with GAP shows that 30,591 of the 34,297 groups of order $5^7$ satisfy this equality; that 7,882 of the 9,310 groups of order $3^7$ satisfy the equality; and that 53,499 of the 56,092 groups of order $2^8$ satisfy the equality.


Here's an easy bit of GAP code to check this:


`glist:=AllSmallGroups(Size,5^7,G->DerivedLength(G) = 1 + LogInt( 
 NilpotencyClassOfGroup(G) , 2));;
 NrSmallGroups(5^7);
 Length(glist);`


Partial information may be attainable, such as constraining various aspects of their character theory, but off-hand I'm not sure. [This paper of Glasby's](https://ac.els-cdn.com/0021869389902044/1-s2.0-0021869389902044-main.pdf?_tid=cdf06e17-bfe3-408f-a679-396a7b42dd10&acdnat=1521790385_037dc746637fc7456996f9daaf0b0326) seems to contain all the references needed for the original derivations of the inequality in question (he references [1],[2], and [4] in his bibliography).


