Final answer:
The early success of the group in question, which would become The Beatles, occurred at the Casbah Coffee Club. This venue was run by the parents of Pete Best, who joined the group as a drummer.
Step-by-step explanation:
In 1959, the group had early success at the Casbah Coffee Club, run by John and Mona Best, whose son, Pete Best, would soon join the band on drums. The Casbah Club was significant in the early days of The Beatles, who at the time were still developing their sound and stage presence. It served as a stepping stone before their later successes in more famous venues like the Cavern Club and their eventual international superstardom.