public school vs Gymnasium

public school and Gymnasium both are nouns.

Word	Noun	Adjective	Verb	Adverb
public school	Yes	No	No	No
Gymnasium	Yes	No	No	No

As nouns, Gymnasium is a hypernym of public school; that is, Gymnasium is a word with a broader meaning than public school:

public school: private independent secondary school in Great Britain supported by endowment and tuition
Gymnasium: a school for students intermediate between elementary school and college; usually grades 9 to 12

Other hypernyms of public school include lycee, lyceum, middle school, secondary school.

public school (noun)	Gymnasium (noun)
a tuition free school in the United States supported by taxes and controlled by a school board	a school for students intermediate between elementary school and college; usually grades 9 to 12
private independent secondary school in Great Britain supported by endowment and tuition