Derived length gives no upper bound on nilpotency class
We first show that for , there exist groups of arbitrarily large nilpotence class.
For , the dihedral group , given by the presentation:
has nilpotence class , but solvable length , since it has an abelian normal subgroup such that the quotient is also an abelian group.
To get an example of a group of length exactly for that has arbitrarily large nilpotence class, take the direct product of with any nilpotent group of solvable length precisely .