(The following is based on my own personal canon, aka fanon, and is not to be taken as true Fallout canon of any kind other than RP scenarios and/or RP fanon/canon. Enjoy.)
Vault 27, dubbed in official Enclave records as The Academy Vault (T.A.V.) is a heavily populated Vault-Tec Vault intended by the Enclave to act as a 'continuity locale' in the event things didn't go as planned for continuity of government. It has held beyond ample manpower reserves for over 200 years after the Great War, and after the failed Enclave campaign in the Capital Wasteland, it has since been fallen back too and the results (rather, it's already-designated results) per it's Societal Preservation Project mission statement, have been plucked.
As part of it's mission statement, this Vault would be deliberately overcrowded. 2000 people were assigned to enter, double the total sustainable amount. It, miraculously, lasted after over 700 (735 exactly) of it's original inhabitants died by smothering to death minutes after the bombs fell. The dwellers, after getting rid of the corpses, had just enough room and supplies to survive. The inhabitants were forbidden from reproducing (The Overseer's family continued itself with specifically chosen dwellers however) until a certain age and only the third generation of the 27 dwellers were permitted to begin bolstering their numbers again regularly. Biodiversity was designated mandatory for 'the sake of purity' as inbreeding apparently went against the Overseer's enclosed ethics guidelines.
Despite the (now classified) sacrifices, the Enclave have designated Vault 27 a notably critical breakthrough of the S.P.P.. It is located in the Columbia Commonwealth in a classified location...