Socle equals Omega-1 of center in nilpotent p-group

Statement

In a nilpotent p-group $G$, the Socle (?), i.e., the subgroup generated by all Minimal normal subgroup (?)s, equals the subgroup $\Omega _{1}(Z(G))$: the first omega subgroup of the center of $G$.

Related facts

• Minimal normal implies contained in Omega-1 of center for nilpotent p-group
• Minimal characteristic implies contained in Omega-1 of center for nilpotent p-group
• Minimal normal implies central in nilpotent
• Minimal characteristic implies central in nilpotent