To go live on your Facebook Group you'll want to click on the cog at the top right of the Lightstream editor to access the stream settings panel. From here you'll want to select the Group that you would like to go live to from the "Go live on" dropdown box.
Note: To go live on a Facebook Group you'll need Admin permissions at the moment.
Once you've selected the group, you'll need to proceed to the group's page on Facebook, click "...More" towards the top middle of the page, and then click on "Edit Group Settings".
Once on the group settings page, scroll down to "Apps", click the Add Apps button and then scroll down to Lightstream to install it to your group