Is there a class of solvable groups $G$ having a derived length $O(\log\lvert G\rvert)$?

See Wikipedia for the definition of Big-Oh ($O$) and the definition of derived series of a group.

Any help would be appreciated. Thank you in advance!

Edit (YCor, after comments below): the intended meaning of big-O is probably not the one linked at, but the "computer science" meaning, namely "$u_n=O(v_n)$" if $c_1v_n<u_n<c_2v_n$ for positive constants $c_1,c_2$ and $n\gg 1$, usually denoted $u_n=\Theta(v_n)$. [The main mathematical use of $u_n=O(v_n)$ being only $u_n<c_2v_n$, which makes the question not interesting.]