I'll just leave a list for you here. All of these do task management, have a contacts list and allow assignment and communication of things between people, and having a free version of some type. I've used all of them at least briefly, and I came away with a positive experience. I currently use Doit.im on a daily basis for myself personally, but communication is not key in my instance. I would highly suggest Podio or Trello for you.
This app is based around GTD, so you have your tasks, contexts, tags, projects and such. Your main screen is your task list, and it's primarily geared towards individuals. In staying consistent with GTD, you can assign tasks to other people. Instant sync across devices requires payment, but it's only $2/month or $20/year, which I Think is quite reasonable. All functionality is present without payment though.
What I really love about Doit.im is the mobile interface. It is really, really easy to add tasks and give them contexts, deadlines, tags, priority, and projects and everything very quickly on the fly. It is also very good with context in terms of "What can I do now". The other apps make the group's life easier, this makes your life easier, and also allows for collaboration. It also integrates with Google Calendar and Google Tasks. Anything with a start or end date on it will be synced to whatever Google Calendar you have listed, so you could sync it to a group calendar so people that aren't even using Doit.im can still see what's going on and what's done and what isn't
Podio is geared towards businesses. It is a collaboration tool meant to replace email and pretty much any and all administration tasks. It is highly extensible, and allows for inclusion of projects, meetings, budgets, timesheets, and basically anything you can imagine. Your main screen is kind of like a facebook feed, as it is collaboration software. You can basically make it whatever you want, and I would suggest checking it out on their website. It's free as long as your group you're collaborating with is only 5 people. Beyond that, it's $8/person a month, which is a bit high, but I don't know of anything else out there that has anything even close to this functionality. If you want collaboration and there's only 5 people in your group, I see no reason to use anything else.
Trello is primarily workflow management software, so it's good for things like software development where tasks have a clear process and you can move tasks across the board as they are completed. It excels at letting you know exactly what is going on, and who is doing it at a glance. It's also completely free, so that's great. This is probably ideal if you're working on a bunch of projects or a lot of people you're working with. I'm currently using it to plan my international move, and like it.
This is similar to Trello in that it's a Kanban type of system that's primarily goal is organizing your workflow. It works very well, and also has integration with The Pomodoro Technique for when you're at your desk or something and need to stay on task, which I really like. Like everything else it has collaboration and mobile apps and such.
Finally, list comparing a bunch of project management software on Wikipedia. The choice really just depends on exactly what kind of implementation you're looking for, as everything is different. I've been through at least 10 different task/project management programs, and there was just one thing with each of them that really bugged me so I switched, but that's kinda how it is.