You cannot programatically fill in login information, that is against the Facebook terms and conditions. You can, however, authenticate as an application as opposed to a user.
Use the following method:
Make a GET request to:
https://graph.facebook.com/oauth/access_token?grant_type=client_credentials&client_id=CLIENT_ID&client_secret=CLIENT_SECRET
Facebook returns:
access_token=SOME_TOKEN
Use this token as your access token and it should allow you to access the group. I have tested this with my application and can confirm it works.
You request the wall information via the request:
https://graph.facebook.com/GROUP_ID/feed?access_token=SOME_TOKEN
This will not pop up any login screens as you are not required to be logged in to view a public group. Ensure your privacy settings are public for the group as well.