If $G$ is a nilpotent group with positive class $c$, its derived length is at most $[\mathrm{log}_2c]+1$.
This statement can be proved by the inclusion of groups in the derived series and central series.
But I don't know how to prove
The class of a nilpotent group cannot be bounded by a function of the derived length.
I think I should find a sequence of nilpotent groups for which the derived lengths are equal, but the classes are not bounded. But I have no idea.
Thanks very much.