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For a group $G$ and a finite set $A$, denote by $\text{End}(A^G)$ the monoidof all continuous shift commuting self-maps of $A^G$ and by $\text{Aut}(A^G)$its group of units. We study the minimal cardinality of a generating set, knownas the rank, of $\text{End}(A^G)$ and $\text{Aut}(A^G)$. In the first part,when $G$ is a finite group, we give upper and lower bounds for the rank of$\text{Aut}(A^G)$ in terms of the number of conjugacy classes of subgroups of$G$. In the second part, we apply our bounds to show that if $G$ has aninfinite descending chain of normal subgroups of finite index, then$\text{End}(A^G)$ is not finitely generated; such is the case for wide classesof infinite groups, such as infinite residually finite or infinite locallygraded groups.

On the minimal number of generators of endomorphism monoids of full shifts