I have the following questions but I don't totally understand them, so im not sure about it all:

Let H be the subgroup of $\displaystyle S_8$ generated by the permutations

$\displaystyle \alpha = (1 2 3 4)(5 6 7 8), \beta = (1 5 3 7)(2 8 4 6)

$

(1) Find the order of $\displaystyle \alpha, \alpha\beta$. What's the order of $\displaystyle \beta^{-1}\alpha\beta$ ?

(2) Find the order of H.

(3) Find the conjugacy classes of H.

(4) Show that $\displaystyle Z=\{e, \alpha^2 \}$ is a normal subgroup of H

(5) To which well-known group is the quotient H/Z isomorphic?

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For (1) I know the orders are all 4 I think.

For (2) Is the fastest way of doing this to actually go through and compute every new element?Or is there an easier way of finding the order?

For (3) I think the conjugacy classes are the sets in which the permutions all have the same cycle structure?

For (4) Z is normal iff Z is a union of conjugacy classes?

For (5) I think this is isomorphic to the Klein Four Group. Find the orders of elements in H/Z ?

Thanks for any help.