How to push Facebook group in google?

Hello guys. I need your advices and recommendations. Please take it seriously. Thank!

I have group in the Facebook, which are ranking with 2 keywords in google. So, my competitor has same group (it is a city news), but he ranking with 6-10 keywords including the name of his site.

So, how can increase the number of keywords, by which my group are ranking in google?

How better to pushing my group by this keywords in google? Classic link building? Or are there other secrets here?

Thank you in advance for all the help!!!

Before anything else: doing SEO for Facebook is not a good idea. You're basically investing your time into promoting FB for free. They can simply close your page and you lose 100%.

Other than that, it's like any other SEO, except you don't control anything in the source code. Add content....

maybe add your city name to the group name

Build some anchor links in reddit and get traffic from high authority websites to the group.

Thank you all for your advice!!!