The capital of the Cocos (Keeling) Islands is West Island. It serves as the main settlement and administrative center for the territory.

The Cocos (Keeling) Islands were discovered in 1609 by Captain William Keeling, but remained uninhabited until the nineteenth century, when they were brought under the control of the British Empire in 1857. On 23 November 1955, the islands were transferred to Australian control under the Cocos (Keeling) Islands Act 1955.

The Cocos Keeling Islands are one of Austtralia's external territories. They are located about 2940 km northwest of Perth, Western Australia. From the islands to Sydney, NSW on the eastern coast is a distance of 5990 km.

Pauline Bunce has written:

'The Cocos - Keeling - Islands'

I found the answer on wikipedia. Not sure if this is the zip code for the airport but this is the zip code for the islands. 6799