Maybe I'm wrong but I wasn't able to find any cost for joining a magical group, in BP at character creation. Am I missing it somewhere?

I was thinking that if there isn't already a cost it would make sense for there to be one. Either as a contact or based on the features of the group.

It costs 5 Karma. Joining a Group is described on page 69 of Street Magic.

Is there a BP cost or is it not a available at character creation?

It is usually not available at chargen, but then again neither is Initiation.

If I were the GM, I would let you be a member of [insert Group in which character is interested], but not necessarily a 'full' member, which would come with the in-game payment of 5 Karma (and meeting all the other requirements).