# Derived length gives no upper bound on nilpotency class

From Groupprops

## Statement

For , there exist Nilpotent group (?)s of Solvable length (?) and arbitrarily large Nilpotence class (?).

## Related facts

### Converse

## Proof

### Dihedral groups

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 .