Although it may be confusing, you could use a subgroup. The main group would be the less restrictive one (chicken), then the one with limited access (bird) would be the subgroup. You'd need to do it this way because you have to be a member of the main group before joining the subgroup. If you reversed it, the 'chicken' members could see all posts. You could limit who would be able to join the subgroup.
I don't think there's a way to 'link' 2 groups the way you want to.
Duane
Duane