Here's another family of examples with arbitrary large $k$, more based on linear algebra. Fix any odd prime $p$. Consider the group $G_{p,s}$ of square matrices of size $s+2$ over the field $F=\mathbf{Z}/p\mathbf{Z}$ of the form $$m^\pm(u,v,z)=\begin{pmatrix}\pm 1 & ^tu & z\\ 0 & I_k & v\\ 0 & 0 & 1\end{pmatrix};$$ with $u,v\in F^s$ and $z\in F$. Its order is $2p^{2k+1}$. Its center is trivial (the center of the subgroup of index 2 is reduced to the cyclic group of elements $m^+(0,0,*)$.

For any $s-1$ elements in $G_{p,s}$. Then they are contained, for some hyperplane $H$ of $F^s$, in the subgroup $\Gamma$ consisting of those $m^\pm(u,v,z)$ with $u\in H$. Then there exists $w\in F^p\smallsetminus\{0\}$ such that $^tuw=0$ for all $u\in H$. Then $m^+(0,w,0)$ belongs to the centralizer of $\Gamma$. Hence $k(G_{p,s})\ge s$ (actually $\le s+1$).

Here's another family of examples with arbitrary large $k$, more based on linear algebra. Fix any odd prime $p$. Consider the group $G_{p,s}$ of square matrices of size $s+2$ over the field $F=\mathbf{Z}/p\mathbf{Z}$ of the form $$m^\pm(u,v,z)=\begin{pmatrix}\pm 1 & ^tu & z\\ 0 & I_s & v\\ 0 & 0 & 1\end{pmatrix};$$ with $u,v\in F^s$ and $z\in F$. Its order is $2p^{2s+1}$. Its center is trivial (the center of the subgroup of index 2 is reduced to the cyclic group of elements $m^+(0,0,*)$.

For any $s-1$ elements in $G_{p,s}$. Then they are contained, for some hyperplane $H$ of $F^s$, in the subgroup $\Gamma$ consisting of those $m^\pm(u,v,z)$ with $u\in H$. Then there exists $w\in F^p\smallsetminus\{0\}$ such that $^tuw=0$ for all $u\in H$. Then $m^+(0,w,0)$ belongs to the centralizer of $\Gamma$. Hence $k(G_{p,s})\ge s$ (actually $\le s+1$).