Is a finite group generated by representatives of its conjugacy classes?
This was asked, and answered, on MathOverflow some time ago: https://mathoverflow.net/questions/26979/generating-a-finite-group-from-elements-in-each-conjugacy-class
Yes. Suppose not: then there will be some maximal subgroup $M\le G$ intersecting each conjugacy class. Then, because $G$ is the union of its conjugacy classes, $G$ is the union of conjugates of $M$. But this is impossible. (Can you see why? Try counting how many elements one can have in the union of $M$ and all its conjugates, noting there are at most $[G:M]$ such conjugates.)