This is a long question, but hopefully someone can give me a suggestion, as I've been hitting my head against the wall...
Take a non-abelian group $P$, of order $p^3$, $p$ prime. We've already proven the following:
-$P$ has $p^2$ characters of degree 1, and $p-1$ irreducible characters of degree $p$. These are all the irreducible characters.
-$P$ has $p^2+p-1$ conjugacy classes. $p$ of these have size 1, $p^2-1$ have size $p$.
I still need to prove that: The classes of size $p$ are the non-identity cosets of the centre of $P$.
And that: given $\chi$, an irreducible character of degree $p$ $\implies$ The representation affording $\chi$ is faithful.
Let $\chi \in \text {Irr}(G)$ with $\chi(1)=p$. Assume that the normal subgroup $\ker(\chi) \neq 1$. Since $G$ is a $p$-group it follows that $\text {ker}(\chi) \cap Z(G) \neq 1$. But $|Z(G)|=p$, hence we must have $Z(G) \subseteq \ker(\chi)$. Since $G$ is non-abelian of order $p^3$, we have $G'=Z(G)$ and it follows that $G' \subseteq \ker(\chi)$, which can only happen if $\chi$ is linear, a contradiction. So $\ker(\chi)=1$, that is $\chi$ is faithful.