I was trying to find the conjugacy classes of $A_5$. So I started by writing out all the conjugacy classes of $S_5$ in the hope that I could just restrict the set of them. The conjugacy class representatives of $S_5$ are $$e,(12),(123),(1234),(12345),(12)(34),(12)(345)$$ So I restricted this set to even permutations to get $$e,(123),(12345),(12)(34)$$ but apparently $(12345)^2$ represents a conjugacy class as well. So I then computed it as $(13524)$. I cannot see why this would be conjugacy class as well. As far as I was aware conjugacy classes in $S_5$ are determined by cycle type, so I thought have thought this holds for the restriction to $A_5$.
Now how would I have known a priori that $(12345)^2$ represents another conjugacy class or that I was missing a conjugacy class (assuming we dont know the number of irreducible representations).
Is there a better way to approach this question?