What is the name of the London club founded by Sherlock Holmes’s brother Mycroft?

The London club associated with Mycroft Holmes is the Diogenes Club.

It appears in “The Adventure of the Bruce-Partington Plans,” where Sherlock Holmes visits the club with Dr Watson. Mycroft is one of its founders and a regular member. The club’s unusual rules require members to avoid speaking to one another, except in the Stranger’s Room, which is reserved for visitors.

The name alludes to Diogenes of Sinope, the ancient Greek philosopher famous for his austere life and disdain for social conventions. Doyle uses the club as a comic contrast to ordinary gentlemen’s clubs: it is a place designed specifically for people who do not want conversation.

The club is fictional, although its atmosphere reflects the private clubs familiar in late-Victorian London. Mycroft’s membership also reinforces his character. He is brilliant, solitary, and comfortable with abstract information, while Sherlock prefers direct interviews, fieldwork, and disguises.