Solution
= Solution
Write an element as $(x,y,z)$, with multiplication
$$
(x,y,z)(x',y',z')=(x+x',y+y',z+z'+xy').
$$
The centre consists of $(0,0,z)$. Each central element is its own conjugacy class. If $(x,y)\ne(0,0)$, conjugation leaves $x,y$ fixed and changes $z$ by every element of $\mathbb F_p$, so its class is
$$
\{(x,y,z):z\in\mathbb F_p\}.
$$
\b[Thus there are $p$ central singleton classes and $p^2-1$ noncentral classes of size $p$.]