Please post an active link starting with http:// to the blog in question.
I don't know is a Group is a Facebook Page or not. I do know this.
If you would like to Publicize your posts to your Facebook page, instead of your Facebook profile, click on the “Set Options” link underneath Facebook on your Settings > Sharing page. You will need to connect your blog to Facebook first. You will need to use a Facebook account that has a Profile with administrative rights to the Page.
Note: A Page-only account can not be used with Publicize. The Facebook account must have a profile. http://en.support.wordpress.com/publicize/#facebook-pages
I also knoe you can use RSS Grafitti http://www.facebook.com/RSS.Graffiti for posting to Facebook walls if you prefer. RSS Graffiti periodically checks the RSS/Atom feeds that you specify and posts any new entries it finds to the Facebook Walls that you specify. You can get any feed written on any wall (Facebook Profiles, Fan Pages, Groups, Events and Application Profile Pages). In fact, multiple feeds to multiple walls. You choose the combination.