In order to go live to a Facebook Group from Livestream Studio, you must first be an Admin and install the Livestream Studio Facebook App to your Group. Once you have done so, you will be able to choose your Group as a streaming destination within Studio.
First, make sure you are an Admin of your Group. Only Admins can adjust Group settings and go live to Groups. Also, ensure that you are using Livestream Studio version 5.1.25.
On your Group's Discussion page, open the More menu and select Edit Group Settings.
Scroll down to the Apps section and select Add Apps.
Find the Livestream Studio app in the list and select it. Once you've added the app to your Group, be sure to scroll to the bottom of the settings screen and click Save.
In the Livestream Studio software, navigate to the Stream tab and select Facebook. Configure your stream as you normally would; your Facebook Group should appear as an option in the Post As menu.