private corporation vs corp

private corporation and corp both are nouns.

Word	Noun	Adjective	Verb	Adverb
private corporation	Yes	No	No	No
corp	Yes	No	No	No

As nouns, corp is a hypernym of private corporation; that is, corp is a word with a broader meaning than private corporation:

private corporation: a corporation owned by a few people; shares have no public market
corp: a business firm whose articles of incorporation have been approved in some state

Other hypernyms of private corporation include corporation.

private corporation (noun)	corp (noun)
a corporation owned by a few people; shares have no public market	a business firm whose articles of incorporation have been approved in some state