Maxson's battlecoat is worn by Elder Arthur Maxson, leader of the East Coast Brotherhood of Steel, and consists of a black BOS officer uniform underneath a heavy padded leather coat with the Brotherhood insignia displayed on the right shoulder. Its already formidable protection can be enhanced even further by upgrading it with the ballistic weave armor mod. While the coat can't be worn underneath additional armor pieces, fully upgraded it offers better protection than the Railroad armored coat V and weighs less. On top of that, it grants a bonus to Perception.
Various terminal entries aboard the Prydwen shed some light on the battlecoat's background and development, even mentioning its considerable weight.
The battlecoat is worn by Elder Maxson. Since Maxson cannot be pickpocketed, using the Pickpocket perk level 4 will not work. The only way to obtain the battlecoat is to kill Maxson and take it from his body.