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, ${\mathrm{\Omega }}_1(Z(G))$ 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, ${\mathrm{\Omega }}_1(Z(G))={\mathrm{\Omega }}_1(G)=2\mathbb{Z}/4\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$. On the other hand, the subgroup $\stackrel{\mathrm{g}}{\mathrm{A}}(G)=2\mathbb{Z}/4\mathbb{Z}$ is a strictly smaller nontrivial characteristic subgroup of $G$.