Major Edgar is a British spymaster.
'Major' here is his rank, not his first name. The man's actual given name is never actually given us. He is addressed, as is common in the military, by his last name and the occasion never arises for him to do more in identification that his rank and surname. The rank designation is added here largely to not have it left ambiguous whether Edgar is the man's first or last name.
When we first meet him, his rank is that of Captain but he will after some time get a promotion and will spend much of the time that we follow him in the higher rank so it is fitting we use that. Throughout the remainder of the War he will not receive any more promotions that we know of.
Edgar possesses, according to one of the agents who will be persuaded to work for him, "one of the most unremarkable faces he'd ever seen. It had the tanned complexion of someone who spent plenty of time outdoors and dark eyes with a penetrating stare, but otherwise there was nothing about it that was memorable. [He] could have stared at it for hours and still had difficulty picking it out of a crowd. The man could have been anything from late-thirties to mid-fifties, and when he spoke it was in grammar-school tones, with perhaps the very slightest trace of a northern accent."
Edgar is described here as a spymaster, not a spy, because it is not his job to go out into the field to collect intelligence but rather he is the man who directs others to do it. How or why he got that job is never really explained but since someone has to do it and he is very obviously quite capable at it, it is easy to see why he was chosen. It should not be taken that Edgar does not ever go into danger because he does, even going into Germany when required but for the most part, Edgar's job is usually more that of director than actor.
Edgar possesses a quite sardonic sense of humor though he is smart enough to keep it low-key especially when dealing with those of higher rank who would likely not get his comments and certainly would not approve of them. In one exchange with an important fellow from the Home Office, Edgar is asked how well he spoke German and he said he was fluent in it, which he was. The man then asked how fluent to which Edgar replies, "Fluent". The bigwig presses that "there's fluent... and fluent" to which Edgar comments, "Perhaps you could explain the difference between fluent and fluent, eh? They sound very much the same to me."