class (n.)
c.1600, "group of students," from French classe (14c.), from Latin classis "a class, a division; army, fleet," especially "any one of the six orders into which Servius Tullius divided the Roman people for the purpose of taxation;" traditionally originally "the people of Rome under arms" (a sense attested in English from 1650s), and thus akin to calare "to call (to arms)," from PIE root *kele- (2) "to shout" (see claim (v.)). In early use in English also in Latin form classis.

School and university sense of "course, lecture" (1650s) is from the notion of a form or lecture reserved to scholars who had attained a certain level. Natural history sense is from 1753. Meaning "a division of society according to status" (upper, lower, etc.) is from 1772. Meaning "high quality" is from 1847. Class-consciousness (1903) is from German klassenbewusst.
class (v.)
1705, "to divide into classes," from class (n.) or French classer. Sense of "to place into a class" is from 1776. Related: Classed; classing.