As a “private social network” platform, I’d really suggest you try Discourse, the forum software where you’re reading this right now. It’s free, open source, modern, high-quality software and together with now several hundred available plugins does not leave many feature wishes unfulfilled.
The “trick” is, I think, to find the right way to use a forum as a kind of social network platform that suits your needs.
Because probably, you don’t really need the “feed-style” social network experience for a small-ish group of people. It’s great for platforms that want to jam in ads between the content, but in terms of networking it rather leads to shallow posts and shallow discussions. You may want to dedicate a part of the forum to this kind of quick, shallow updates, as sometimes relevant discussions can develop from that. That’s what @johncoate introduced here, by means of the long-running “status update” threads.
And neither would you need the follow / friends features for a small-ish social network. It’s better to not compartmentalize a small group but to have them all see the same content (which may be private if desired, as Discourse allows that).
There are several more features in Discourse that allow you to tweak the forum experience to what suits your situation. Just some, off the top of my mind:
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mailing list mode, to provide an additional old-school interface in case you have many people there from the old days of the web, where discussions often happened on mailing lists
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curated blog, possible by means of using tags on the relevant topics to group them together, and also installing the topics preview plugin so that an address like
https://example.com/tags/blog?view=topic-preview
then shows a blog-style view -
the Discourse Follow plugin, if you want a follower feature where people can get notifications about the content and actions of people they follow
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to allow adding additional multimedia content, you can tweak the Onelink Embed system; it also offers more options than is obvious, for example it’s out-of-the-box possible to embed 360° images (example: third picture in this post)