In this list of review questions, there is the following question about $GL_3(F_p)$.
Question 1.38. Let $G$ be the group of invertible 3 × 3 matrices over $F_p$, for $p$ prime. What does basic group theory tell us about $G$? How many conjugates does a Sylow $p$-subgroup have? Give a matrix form for the elements in this subgroup. Explain the conjugacy in terms of eigenvalues and eigenvectors. Give a matrix form for the normaliser of the Sylow $p$-subgroup.
Most of this is pretty standard. See here, for example. One Sylow $p$-group is the group of upper triangular matrices with $1$s on the diagonal, and conjugating this gives all the others.
However, what does it mean to "Explain the conjugacy in terms of eigenvalues and eigenvectors"? I'm unsure what the question is getting at here. How does knowing about the eigenvectors and eigenvalues shed light on the conjugation process? We could, for example, using rational canonical form and our knowledge of the characteristic polynomial ($(x-1)^3$) to give the similarity classes of matrices in the group, but I don't think that is what was intended. Is there anything interesting to be said about eigenvalues, eigenvectors, and there relation to conjugation?
Perhaps what was meant is that two matrices in this subgroup are conjugate if and only if they have the same number of linearly independent eigenvectors, which can be seen by the fact that this determines the "nilpotent part" of Jordan canonical form.