Find the conjugacy classes of $A_5$ 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? 
 A: The conjugacy classes of $A_5$ are the orbits of the action of $A_5$ in $A_5$ given by the conjugacy action.
You know that in $S_5$ everyone of those elements are in an unique conjugacy class and they represent all the classes. You know from the class equation that (in $S_5$)$$|\mathrm{orb}(x)_{S_5}|=\frac{|S_5|}{|\mathrm{Stab}(x)_{S_5}|}$$
We are trying to study $\mathrm{orb}(x)_{A_5}$.
Since $\mathrm{Stab}(x)_{S_5} < S_5$, we have two possibilities:
1) $\mathrm{Stab}(x)_{S_5} \subseteq A_5$: in this case $|\mathrm{orb}(x)_{A_5}|=\frac{1}{2}|\mathrm{orb}(x)_{S_5}|$, so your class in $S_5$ splits in two new classes in $A_5$.
2) $\mathrm{Stab}(x)_{S_5} \nsubseteq A_5$: since $A_5$ is a subgroup of index 2, and since $$\mathrm{Stab}(x)_{A_5}=A_5 \cap \mathrm{Stab}(x)_{S_5}$$ you get$$[\mathrm{Stab}(x)_{A_5}:\mathrm{Stab}(x)_{S_5}]=2$$
So you have $|\mathrm{orb}(x)_{A_5}|=|\mathrm{orb}(x)_{S_5}|$, and you get the same conjugacy class.
Moral: you just have to know if $\exists \tau \in \mathrm{Stab}(x)_{S_5} | \tau \notin A_5$ i.e.: if we are interested in studying if the conjugacy class of $(123)$ splits when you go in $A_5$, bacause $(45) \in \mathrm{Stab}((123))_{S_5} \smallsetminus A_5 $ you know that it DOES NOT SPLIT. 
Was I clear?
