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If $𝒳$ is a class of groups, then a group $G$ is said to be minimal non $𝒳$-group if all its proper subgroups are in the class $𝒳$, but $G$ itself is not an $𝒳$-group. The main result of this note is that if $c\>0$ is an integer and if $G$ is a minimal non $\left(\mathrm{ℒℱ}\right)𝒩$ (respectively, $\left(\mathrm{ℒℱ}\right){𝒩}_c$)-group, then $G$ is a finitely generated perfect group which has no non-trivial finite factor and such that $G/Frat\left(G\right)$ is an infinite simple group; where $𝒩$ (respectively, ${𝒩}_c$, $\mathrm{ℒℱ}$) denotes the class of nilpotent (respectively, nilpotent of class at most $c$, locally...