The number of elements of a conjugacy class of an element is equal to the index of the centralizer of the element.
Look at the elements which we did not use yet, for example . Now contains . Thus, . By Lagrange's theorem this means since it means . Thus, which means the number of elements in the conjugacy class of is . Conjugating by we get . Thus, is another conjugacy class. Repeating the same argument with we find that (since at least contains . Thus, the number of element in the conjugacy class of is . Conjugating by (it does not have to be in fact it can be any element) we get . Thus, is another conjugacy class. Reapting the same argument we find as another conjugacy class. But we are still left with . However, commutes with . Therefore, and it is in its own conjugacy class.
The conjugacy classes are: .
To find normal subgroups of order 8 it is sufficient to find subgroups of order 8 because since they are index two they are automatically normal. It actually has three subgroups of order 8. But the two more obvious ones are: , i.e. a copy of and .
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