Let $G$ be a finite group and $p$ be an odd prime. It's known by work of Quillen that the minimal primes of $H^{2\ast}(G;\mathbb{F}_p)$ are in one-to-one correspondence with the maximal elementary abelian p-subgroups of $G$. Explicitely, if $E \le G$ is maximal elementary abelian, than the corresponding minimal prime is

$$\mathfrak{p}_E = \operatorname{ker}\lbrace\hspace{1pt} H^{2\ast}(G;\mathbb{F}_p) \to H^{2\ast}(E;\mathbb{F}_p) \to H^{2\ast}(E;\mathbb{F}_p)/\sqrt{0} \hspace{1pt}\rbrace$$

where the first map is restriction and the second is the natural epimorphism. Since $H^{2\ast}(G;\mathbb{F}_p)$ is noetherian, minimal primes are associated. Therefore, I wonder:

Is there an explicit description of an element $x \in H^{2\ast}(G;\mathbb{F}_p)$ such that $\mathfrak{p}_E = \operatorname{Ann}(x)$ ?

By "explicit" I also mean transfers, power operations, chern classes, etc. of some element that can be described in a concrete manner.