Consider $G=GL_n(\mathbb{Z}/2\mathbb{Z})$. What is the smallest $n$ for which $G$ has an element of order 8? Give an example of an element of order 8.
I've thought about just considering what happens to powers of Jordan blocks since each matrix in $G$ is similar to a matrix in Jordan normal form. I've managed to convince myself that the smallest $n$ is 4, but I have no idea how to go about finding a specific element of order 8. Is there a better way than guess and check? There are far too many elements in $GL_4(\mathbb{Z}/2\mathbb{Z})$ just to guess and check.