I'm a member of many groups because of the freebies I get from Gleam. I don't want any notifications or their feeds in my friend's activity while being the part of group.
So i dont know if the solution i can give you also results in the Steam Client not giving you notifications from groups anymore.
I did this in the Browser:
If you are on your activity page on the top of the list there is a link called Friend Activity Settings:
Now you should see a lot of options and checkboxes. Scroll down until you find the groupbox A Group i belong to:
You can change the settings now and save it on the bottom of the page.
As i said above i dont know if this is just for the activity feed or also for the steam client (which theoretically would be logic).