class's = owned by one class. classes's = owned by multiple classes. (usually shortened to classes')
You can't shorter class's so class', however. Here's why: <em>You can only shorten it when it's a plural noun that ends in an s or when it's a proper noun (names, places) that ends in an s. </em><em> </em>So unless there's more than one class, it's class's.<em> </em>