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

Statement
In a nilpotent p-group $$G$$, the fact about::socle, i.e., the subgroup generated by all fact about::minimal normal subgroups, 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