Regarding corporate travel, I take it you already know about the onbusiness scheme?

www.ba.com/onbusiness

Regarding route discounts etc, I'd imagine it would need to be a reasonable amount of flying to cover the work involved in setting it up - as a minimum. No idea though.

Regarding group travel - for business or leisure reasons - again, no idea myself, however I'd guess the 'entry level' barrier to gain something here might be lower and easier than for a corporate discount.