Is it legal to refuse to let students take the Regents as a punishment?

My private school has this rule that if you come late you have to stay for detention. When finals and regents come they say that you have to pay it all off at a rate of $4.00 an hour. This makes some people owe up to $200.00 if they had some kind of family tragedy or something and forgot to bring a note. Anyway, their rule says that if you don't sit all of your detention or pay it all off then you can't take the Regents. As far as I know this is illegal. If anybody can verify this by telling me where it says it in law that would be great. So, can a school refuse to let students take the Regents (or any other state mandated test) as a punishment?