possible class equation for a group of order 8
Solution 1:
$1+1$ says that center contains exactly two elements, say $\{1,z\}$.
Appearance of $4$ says, there is an element $x$ with $4$ conjugates, hence its centralizer is of size $8/4=2$.
But, $x$ commutes with itself, as well as central elements $1$ and $z$; so its centralizer should be of size $>2$.
This is a contradiction!