Owning in a group is generally easier on the finances and a good way to learn. Sharing the costs helps a lot. Managing a plane is something some people enjoy, and others don’t so that can be a benefit as well.
Unfortunately, the caveats are long and scary. I will hit a few.
Your partners, are like wives and children. You become married to them. They crash, you get involved. You need to have legal counsel go over the agreement.
You need to get a good idea of their compatibility on maintenance issues and flying use with you. Some mixes of people work well, others not. Most important is rules in the agreement on deferring maintenance and upgrades. Many partnerships have become nightmares because of incompatibilities and plane bad behavior. Other things to look at are industry insiders using your plane for their profit.
Buying a part of a plane is still buying a plane. Get a pre-buy inspection. This is NOT an annual inspection and should not cost that much (something around $500). On a 172 you are looking for things like corrosion and other things that ANNUALS DO NOT CATCH. Assume airworthiness (usually), you want the inspection on monetary gotchas and commonly missed items. Start with the logs, without even seeing the plane these can stop the deal. Always use someone OTHER than the guy normally used for maintenance.
Usually, its best to limit your partnership to no more than 4 parties. More than that means availibility may be worse than renting. Also, the 5th guy only lowers your variable cost by a small margin, and also doesn’t lower your capital costs by enough to warrant the risk and hassles unless the plane is rather expensive.
Now, directly to your situation.
Renting out a plane is a tough business and full of traps. You need legal counsel familiar with both aviation tax and FAR issues. The cost of that is hard to offset with a 172.
If the instructor in the group is giving instruction to people outside the group then this ends up being a rental plane according to the FAR’s (You are now a de facto flight school with all the costs and hassles thereof). It will have higher insurance and maintenance costs. If it’s not making a profit, then why do it? If it is making a profit, why do they want more partners? There may be a logical reason, but if I were a member, I would try to buy up more of a money maker.
Without more specifics from you, that’s about all I can tell you for now.