77th (East Middlesex) Regiment of Foot
This regiment was first raised in 1787, and was named for East Middlesex in 1803. After serving with many casualties in the Crimean War from 1854-56, the Regiment returned to Britain then embarked for New South Wales in 1857. The coatee above belonged to an officer killed during the Crimean, so it did not travel to Australia, but bears the same buttons with the number 77 and the Prince of Wales plumes. They were needed in India, and left the following year.
See also http://www.austbuttonhistory.com/uncategorized/20th-november-2020/
At least one button bearing 77 but without plumes has been found at Victoria Barracks, Sydney.