I am looking for a small (say, finite and of small cardinality) subgroup of the general linear group whose centralizer consists only of scalar matrices.
I work over complex numbers.
A more precise statement of the problem is the following: let $GL_n$ be the general linear group of invertible $n\times n$ matrices. I am looking for $G \subseteq GL_n$ such that $C_{GL_n}(G) = Z(GL_n)$ (where $C_{GL_n}$ is the centralizer and $Z$ is the center), and I would like to know if the smallest such $G$ is known.
This question can be rephrased, in terms of representation theory: if $GL_n = GL(V)$ for a vector space of dimension $n$, I am looking for the smallest $G \subseteq GL(V)$ such that the only $G$-invariant in $V^* \otimes V$ is the identity (up to scale).
I know it can be done with $G = \mathbb{Z}_2 \wr \mathbb{Z}_n$, where $\mathbb{Z_n}$ acts on $n$ copies of $\mathbb{Z_2}$ by cyclically permuting the factors. $\mathbb{Z}_2^n$ embeds in $GL_n$ as diagonal matrices with entries $\pm 1$ and $\mathbb{Z}_n$ embeds as the group generated by the permutation matrix corresponding to the long $n$-cycle ($1$ in the $n-1$ entries immediately above the diagonal and in the bottom left entry and $0$ elsewhere).
This gives a copy of $G$ in $GL_n$ that is centralized only by scalar matrices and its order is $\vert G \vert = n2^n$.
I would bet that it is possible to do better (namely a smaller $G$), maybe even a lot better; but I am not sure what a systematic approach to find such small $G$ could be.