HMC membership - otherwise there'd be no such thing as "a minor public school" (all those in the act being non-minor) and really, that's rather a useful phrase. ;-)

I have a number of exes who went to 'a minor public school' and I am familiar with schools in this area, and I would be comfortable saying that e.g. Abingdon School is a minor public school. It has something more 'public school' about it than other private schools around here - in teaching approach, in facilities and resources, in the behaviour of its pupils...