Nilpotent group with torsion-free abelianization need not be torsion-free

Statement

It is possible to have a nilpotent group $G$ such that the abelianization of $G$ is a torsion-free abelian group but such that $G$ is not torsion-free.

Proof

Example of the integers mod 2 in the center

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Rationally powered abelianization example

See Nilpotent group with rationally powered abelianization need not be rationally powered.