The Octagon was a facility located in Capitol City. It was in the process of being built since parts of the building were unfinished.
The Octagon was based off the Pentagon which monitors activities and military actions. Crpyto first went go to the Octagon when Silhouette ordered Armquist to meet with the Joint Chiefs of Staff in order to unite the Military. Crypto scanned a Scientist to get the password in order to get into the base. The scientist said the password was "Blue Rider". Once Crypto got in, Silhouette and Armquist met in front of the Octagon. After Crypto listened to their conversation, he had to get back to the Saucer and go to Union Town and kill Armquist before he united the Military against Crypto.
That was the area where Crypto battled Silhouette. After Crypto defeated Silhouette, Majestic was finished and Crypto won the game.