Well, when we say community, we refer to a large number of people or entities, who may be affiliated to many smaller or disparate sub-structures.Community almost always refers to a varied and large audience, yet with certain things, which maybe certain interests, opinions, or religion or ethnicity.
On the other hand, group is indeed most often used for a small number of people or other entities, each of which may or may not be large. Also, a group is not as natural a collection as community. A group maybe formed by some of us just now, but a community arises on its own and comes together, and has more naturally common attributes. Similarly, one can be expelled from a group, if he disrespects the rules, but one cannot be banned from a community per se, unless it is something very artificial like a virtual social networking aggregate that is moderated etc.
E.g. we talk about the international community of researchers of a certain field, say Physics, or the international community of institutional investors (each of which is an institution like a bank and consists of millions of people), whereas we say G8- a group of nations.
It means based on similar conditions or principles, or in a state in which a comparison can be made.
For example, an employer tells his employee he will be paid time and a half for every hour he works after nine o'clock and double for every hour he works after midnight. He then tells another employee he will be paid on a comparable basis, i.e. the employee will have his hourly pay determined in a similar way.