The Germans tried several 50 mm and 55 mm AAA designs, with a notable lack of success.
The nominal effective ceiling of 37 mm weapons was about 4-4500 m, but in reality it was closer to 2500 m with visual aiming. This created a AAA 'deadzone' - in reality, just a less than optimal zone for the light AA/heavy AA crossover, between 3000 and 4500 m where Allied medium bombers could operate.
Theoretically, a 50 mm class weapon was the solution to this, with a proper effective range of about 3500-4200 m, but the designs never quite worked right. The 5 cm Flak 41 was slow firing and heavy, and the ammunition was ill-suited to its requirements, with a smallish amount of HE filler and a tendency to tumble at high velocity.
The 55 mm Gerat 58 derivative was started with this in mind. They upped the M/V to 1050 m/sec and increased the HE capacity of the round by about 20%. The rate of fire suffered with the bigger weapon. With the five round clips used (which weighed over 30 kg each), the gun fired at about 60-70 rpm. This wasn't too bad though, as the single barrel 37 mm/40 mm class weapons really only had practical RoFs ranging from 60 to 120 rpm.
The Gerat 58 wasn't fully finished by the time the war ended. The actual design itself was reasonably solid though. It formed the basis for the post-war Soviet S-60 57 mm gun, which was a very successful design.
The 5.5 cm flak was reckoned to have an effective range of about 4000 m with visual aiming, about 5200 m when radar controlled. The weapon was very powerful, with a M/V of just over 1000 m/sec (compared to around 800 m/sec for the 37 mm flak and 880 m/sec for the Bofors).