Omega-1 of center not is minimal characteristic

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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.
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 $G$.