Yes, Henry VIII ordered the arrest and execution of the Duke of Buckingham, Edward Stafford, in 1521. However, it is unclear whether Henry VIII personally carried out the killing or if it was done at his direct command.

Barbara J. Harris has written:

'Edward Stafford, third duke of Buckingham, 1478-1521' -- subject(s): Politics and government, Social life and customs, Great Britain, Nobility, Biography

William Winde built the original Buckingham Palace, originally called Buckingham House, in 1703. Buckingham House was enlarged by two architects, John Nash and Edward Blore, in the 19th century to become the Palace.

It was built in 1705 as a private residence for the Duke of Buckingham. The title Duke of Buckingham was created in 1444 but became extinct after the execution of the 3rd Duke in 1521 (my guess is that's who you are referencing in your question). It was recreated in 1623 and a few more times after that.

William Winde built the original Buckingham Palace, originally called Buckingham House, as a large townhouse for the Duke of Buckingham in 1703. Buckingham House was enlarged by two architects, John Nash and Edward Blore, in the 19th century to become the Palace.