I will use the fact that the smallest non-abelian group has order 6, which is not hard to show.
Let
be the conjugacy classes of
. Assume WLOG that the first
are non-central.
Suppose
Then
Thus
We will reach a contradiction for each case.
The case
is impossible since this would imply
were abelian. So assume
.
Let
with
. Let
with
. Consider
Since
and
are both noncentral and
has order two we know that
Thus
Hence, we have shown that
for all
But this implies
which is a contradiction.
I don't have time to complete the last case. But it is fairly similar. Again use the fact that we know the group structure of
and find the similar contradiction. Then use the class equation to get the inequality for
. I have not proven the case of equality but I'll leave that for someone else.