First, try to find a partition of 6 using the method shown in the link. You can also find some values of partition function in here. Each partition of 6 corresponds to a conjugacy class in S_6.

For instance, the partitions of 6 can be described as,

1. 6 : example cycle type (1, 2, 3, 4 ,5 ,6)

2. 5+1 : example cycle type (1, 2, 3, ,4 ,5)(6) = (1, 2, 3, 4 ,5)

3. 4+2 : example cycle type (1, 2, 3, 4)(5,6)

...

Since P(6)=11, there are 11 conjugacy classes in S_6. Note that each cycle type in S_6 represents a conjugacy class(partition) in S_6.