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 ($O$)][1] and the definition of [derived series of a group][2].

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 "$u_n=O(v_n)$" 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.]* 


  [1]: https://en.wikipedia.org/wiki/Big_O_notation
  [2]: https://groupprops.subwiki.org/wiki/Derived_series