Great breakdown how a group on CT differs from a page.
I am pleased to announce that group notifications will now work! If you are in a group, and somebody else posts in the group, you should get both a notification in the notification bell and a 'push' notification.
This applies to the desktop only. Push notifications ARE now happening for the android app, which is new TODAY. However, there are some quirks, which we will work on ironing out next week. To make use of those push notifications for android, go to the app settings, scroll down to the 'notifications' option, and select the options you desire. Save/go back. Then, close the app completely, making sure it isn't running, and then re-start it.
To help people understand the differences between groups and pages I have created this chart.