To emphasize what evaki explained. Here are some things to check.
Make sure you have the Pages "Advanced Options" set for the group:
If you just select Pages, the group will only have "View" permissions.
Here are the minimum Advanced Settings to be able to "Add" and "Edit Content":
If you like, you can add the group permissions to a page, when you create (Add) it:
If you want to add group permissions to a page after it's been created, or if you want to change group permissions for a page, you can do so on the "Change Settings" tab of the Page:
Hope this helps.