No, they are not synonyms.
a synonym of class is room
Class mate
Class, level, or layer.
Rich or upper-class
synonym I had this for class.
-- subject -- theme
Capitalist middle-class, traditional, conventional, materialistic, hidebound
level, class, section, division, department
A synonym for social class could be "socioeconomic status" or "social stratum."
In social studies, a synonym for different could be diverse or varied.
class, order, rank, status, stratum, social order
class, type, kind, genus, sort, variety, order