Omega-1 of center not is minimal characteristic

This article gives the statement, and possibly proof, of the fact that for a group, the subgroup obtained by applying a given subgroup-defining function (i.e., Omega-1 of center) does not always satisfy a particular subgroup property (i.e., minimal characteristic subgroup)
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Statement

Let $p$ be a prime number and $G$ be a nilpotent p-group. Then, $\Omega _{1}(Z(G))$ is not necessarily a minimal characteristic subgroup.

Proof

Example of an Abelian group

Consider $G=\mathbb {Z} /4\mathbb {Z} \times \mathbb {Z} /2\mathbb {Z}$. Then, $\Omega _{1}(Z(G))=\Omega _{1}(G)=2\mathbb {Z} /4\mathbb {Z} \times \mathbb {Z} /2\mathbb {Z}$. On the other hand, the subgroup $\operatorname {Agemo} ^{1}(G)=2\mathbb {Z} /4\mathbb {Z}$ is a strictly smaller nontrivial characteristic subgroup of $G$.