Hi,

First, you seem a bit confused about conjugacy classes in a group. Given a group G and g an element of G, the conjugacy class of g in G is the set cl(g) = {x^{-1}gx : x is in G}. A standard result is the number of elements in cl(g) is the index of the centralizer C(g) in G.

Next, do you know any more about the group D? Or is that part of the problem? To find the conjugacy class of a^{2}in D, you need to know how products are formed in D. In particular, you need to know the element b^{-1}ab. With the information given, (D is a subgroup of S_{6}and D has 12 elements), it is possible to deduce that b^{-1}ab = a^{-1}and that the order of C(a^{2}) in D is 6; i.e. the index of the centralizer is 2. So cl(a^{2}) = {a^{2}, a^{-2}}.