The group $\mathrm{GL}(n,\mathbb{Z})$ acts on $(\mathbb{Z}/2\mathbb{Z})^n$ by right multiplication (the same kind of things can be done with left action). I denote by $H\subset \mathrm{GL}(n,\mathbb{Z})$ the isotropy group of $(1,\dots,1)$, which corresponds to the group of matrices such that each column has an odd number of odd values.

Then $H$ has index $2^n-1$ is maximal and if $n\ge 3$, it is generated by permutations, diagonal matrices and two simple upper triangular matrices with two ones or one two outside of the diagonal.

I came to this group with questions of algebraic geometry and my questions are the following:

Is this group studied somewhere in the literature? Is this useful to something else? Are the (easy) results about generators stated above written somewehere?