The second step that I would do is choose an element not from the ones chosen, say

. Now we will use the theorem which will make computations a little easier. The number of elements conjugate with

has to be

. Now

thus it has

__at least__ four elements, by Lagrange's theorem it can therefore have

elements it cannot be

since

is not in the center thus

thus

which means the number of elements conjugate with

is just two. Now

thus

is conjugate to

,

__we can stop here__ by the theorem thus

is another conjugacy class.

The third step that I would do is choose an element not from the ones chosen, say

. Now we will use the theorem which will make computations a little easier. The number of elements conjugate with

has to be

. Now

thus it has

__at least__ three elements it cannot have all eight thus by Lagrange's theorem

which implies that

. Now

thus

is conjugate to

,

__we can stop here__ by the theorem thus

is another conjugacy class.