It seems like you're asking for an auto-increment that maintains a separate increment per distinct value in another column.
As @RickJames answers, this feature exists in the MyISAM storage engine, but it doesn't work in InnoDB. The reason is related to InnoDB's row-level locking: in order to prevent a race condition with another concurrent client inserting to the same 'class' (per your example), InnoDB would have to create a gap lock on all the rows for the given class.
This doesn't affect MyISAM, which does table-level locking anyway, but for InnoDB it would greatly harm throughput during concurrent workloads.
This is not a good reason to use MyISAM.
But you should abandon the idea that an auto-increment is some kind of ordinal or row number—it isn't. All it is is a unique value; it has no other meaning.
If you need to assign consecutive student id's per class, then do so in your application and insert values explicitly. Don't rely on the auto-increment mechanism to do that.
