A new method for determining genus of a group is described. It involves first getting a bound on the sizes of the generating set for which the corresponding Cayley graph could have smaller genus. The allowable generating sets are then examined by methods of computing average face sizes and by voltage graph techniques to find the best embeddings. This method is used to show that genus of the symmetric group S 5 {S_5} is equal to four. The voltage graph method is used to exhibit two new embeddings for symmetric groups on even number of elements. These embeddings give us a better upper bound than that previously given by A. T. White.