Derived length gives no upper bound on nilpotency class

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Statement

For l > 1, there exist Nilpotent group (?)s of Solvable length (?) l and arbitrarily large Nilpotence class (?).

Related facts

Converse

Proof

Dihedral groups

We first show that for l = 2, there exist groups of arbitrarily large nilpotence class.

For n \ge 3, the dihedral group D_{2^n}, given by the presentation:

\langle a,x \mid a^{2^{n-1}} = x^2 = e, xax = a^{-1} \rangle,

has nilpotence class n - 1, but solvable length 2, since it has an abelian normal subgroup \langle a \rangle such that the quotient is also an abelian group.

To get an example of a group of length exactly l for l > 2 that has arbitrarily large nilpotence class, take the direct product of D_{2^n} with any nilpotent group of solvable length precisely l.