Does someone know of any work done regarding the connection between the derived series & the frattini series of a pro-p group/p -group ?

[ I'm aware of the general fact that the frattini-series can be regarded as the p-derived series ]

What about some connection between central series and derived ?

any varation on this theme will be also appreciated!

Does someone know of any results regarding estimating number of groups in a certain class by using these series?

Thanks in advance !