Conjugacy classes of a non-abelian group of order $p^3$

Question

Let $G$ be a nonabelian group of order $p^3$, where $p$ is a prime. It's well known that $|Z(G)|=p$. The noncentral elements have all centralizer of order $p^2$, because $Z(G)<C_G(x)<G$ for every $x\in G\setminus Z(G)$. Therefore, the noncentral conjugacy classes have all size $p$ (orbit-stabilizer). Say $k$ and $l$ the number of the noncentral conjugacy classes of elements of order $p$ and $p^2$, respectively. Therefore (class equation): $$p^3=p+kp+lp$$ namely: $$k+l=p^2-1\tag1$$ I'm looking for a second equation in the unknowns $k,l$ to match with $(1)$, so as to determine $k$ and $l$ separately.