I'm not an expert or from the museum but i can tell you your find is fossil coral.
I thought we cold get experts to view our finds here. I think you are right, they must have been very thin corals though!
There are plenty of experts on this forum but the thing is that no one is an expert on every type of fossil.
You really dont need to be a coral expert to make just a general i.d, unless you require the exact genus & species that is.
What you could do now is make sure that iam correct & do a little research of your own into fossil corals.
Prehaps starting with carboniferous corals.
By the way if you got that cut & polished it could look really fantastic.
Heres a link that you may find interesting