The primary purpose of Xavier's school from what I have seen is to train mutants to control their abilities. The schooling was for legal reasons since most are teenagers and are required to go to school. X-men Evolution tv show even had Xavier's students attend a public school.

My character which I have no clue about it, didn't manifest any mutant abilities until literally minutes after he/she graduated from high school.

I just might go with a student once I start writing it.