Do you want precision speed or just synchronised to the strobe lamp.
There are no readily available motors with this speed. You are looking at a gearbox. I recomend you get a motor with gearbox fitted.
Precision speed can be achieved with a synchronous motor, which is locked to the mains frequency, or by a feedback control system with a tachometer to measure the speed so the error can be corrected. This approach is likely to be expensive, not what you want. Ordinary induction motors trail the mains frequency with a slip that is dependant on load, but they just might me stable enough for your needs. The same applies to a DC motor. It just might be stable enough.
If the strobe is flashing under its own control, even though the turntable revolves absolutely steadily, the strobe may drift on its own, and it flashes at a random location (rather than synchronised).
The alternative is to trigger the strobe from the turntable. The stability of the turntable is much less important. This is much cheaper to do and provides "perfect" synchronisation. You would need to make up some circuitry yourself, depending how the strobe is triggered. I suggest using strong magnet/s like the rare earth type, mounted on the table edge (I assume it is round). Use a fixed (stationary) logic level hall effect device (momentay action) as the simplest to get going.
The hall effect device changes state (switches on then off again) as the magnet passes within a couple of mm. The orientation of magnet is important. This is used to trigger the strobe. You need to find out how to connect these two together but it is quite possible there is no additional components required. You may need to get help from someone.
Another method is to have bumps or cams on the table that can operate a microswitch. The best microswitch is the type with a lever and a roller on the lever for this case. The table will need to be precisely round, or the cams adjusted individually for the same position from the centre.