Let $G$ be a non-abelian finite $p$-group such that $G/[G,G]$ is elementary abelian group of order at least $p^3$. Let $S=\{\bar{x_1},\bar{x_2},\bar{x}_3,\cdots,\bar{x}_n\}$ be independent generators (minimal generating set) of $G/[G,G]$. Then (we know) that $x_1,x_2,\cdots, x_n$ generate $G$.
Suppose $S_0=\{x_{i_1},x_{i_2},\cdots, x_{i_r}\}$ be a subset of generators of $G$, with $|S_0|\geq 2$. Suppose that, these generators in $S_0$ commute with each other.
Question: Is the subgroup $\langle x_{i_1},x_{i_2},\cdots, x_{i_r} \rangle$ an abelian subgroup of order $|[G,G]|.p^{i_r}$?
Recall: An element $t$ of a group $G$ is called a non-generator if, whenever a set $U\subseteq G$ together with $t$ generates $G$, $U$ generates $G$.
The confusion is that the elements of $[G,G]$ are non-generators of $G$; but I do not know, whether these are non-generators of the subgroup $\langle x_{i_1},x_{i_2},\cdots, x_{i_r} \rangle$ also? Also, the subgroup in question should be abelian; but I want to know, whether stated order is correct? (I have assumed that $G/[G,G]$ is finite elementary abelian $p$-group.