I ride with Chorlton Velo who ride from Didsbury in South Manchester. There’s a few guys who come out from the centre, it’s not too far. We don’t run any chaingangs at the moment but the fast rides can be very fast 
Friendly bunch and I would recommend. You can find them on British Cycling - we ask that people do an introductory Saturday ride before joining the other rides. Organised rides are Saturday, Sunday and Wednesday and split into Social, Intermediate and Fast - the Fast group has quite a few guys who race and are very quick.
I am not very familiar with groups in the city centre, I would have guessed Manchester Wheelers but I don’t see them around much these days.