For the mod $2$ cohomology, see Section 11 of Benson and Carlson, "Diagrammatic methods for modular representations and cohomology" Comm in Alg 15 (1987), 53-121 for the ring structure. If you want to know about the $A_\infty$ structure, and much more information besides, see https://arxiv.org/abs/2208.07913. At odd primes, the group has cyclic Sylow subgroup, so the cohomology is easy.
In the mod 2 cohomology, as given in the first reference above, there's a generator in degree 2, two generators in degree 3, and one relation, saying that the product of the two degree 3 generators is equal to zero. This gives a generating function for the dimensions of $\frac{1+t^3}{(1-t^2)(1-t^3)}$.