Sorry but I agree with PVP.
My advice would be to go down to the gym, arrange to talk to the manager over a cup of coffee, invite him up to your flat when said early morning class is happening to hear the level of noise so he knows you're not being fussy and take it from there.
The gym probably has enough problems surviving against the competition of FH Pools, I think the last thing they need is an official complaint about noise levels.