Yes, getting invited to a group board is the only way to get in, unfortunately.
Many group boards will have instructions in the description on what to do / who to contact to get invited, so just follow those instructions. Usually there will be an email you can sent an invite request to, or they'll tell you to leave a comment on one of the posts in the group.
If there are no instructions, you can try messaging the board creator. The creator's username is in the url of the page, and it is also the first in the list of board contributors at the top left hand side of the group board page. Unfortunately, you can't direct message someone unless they follow you back, so you'll have to try leaving them a comment on one of their most recent posts. It can be difficult getting a reply sometimes, especially if the board has been around for a long time, but it's worth a try so keep at it and don't get discouraged!