Largest order of an element in $\mathbb{Z}_{128}^*$

Question

Unlike $\mathbb{Z}_{144}^*$ we can break them into two group like $\mathbb{Z}_{16}^*$ and $\mathbb{Z}_{9}^*$. and take there largest order of element in group $\mathbb{Z}_{16}^*$ and $\mathbb{Z}_{9}^*$ and take the lcm of it. However, $\mathbb{Z}_{128}^*$ cannot break down into two different groups. So how do we find the largest order of element in $\mathbb{Z}_{128}^*$ with $\phi(2^7)$?

Answer 1

For $n\ge 2$, $\Bbb Z_{2^n}^*$ is the direct product of a cyclic group of order $2$ generated by $-1$ and a cyclic group of order $2^{n-2}$ generated by $5$.

To see $5$ has order $2^{n-2}$ modulo $2^n$, one can inductively prove the congruence $$5^{2^k}\equiv1+2^{k+2}\pmod{2^{k+3}}.$$