It's a very well known result by Gauss that $(\mathbb{Z}/2^n \mathbb{Z})^\times = \langle -1 \rangle \times \langle 3 \rangle \cong C_2 \times C_{2^{n-2}}$. Consider a faithful action $\mathrm{mul}: C_{2^{n-2}} \to \mathrm{Sym}(\mathbb{F}_2^n)$ that is obtained by representing each element of $\m...