Omega-1 of center not is minimal characteristic

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.

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 $$G$$.