Number of groups of a given order?

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Is there a formula for determining the number of different groups up to isomorphism for a group of a given order?
 
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There isn't a general formula, but the number of groups has been tabulated for a large number of values.

http://people.csse.uwa.edu.au/gordon/remote/cubcay/ has a list of the number of groups up to order 1000. An interesting error causes it to say the number of groups of order 512 is -1, but it is actually 10,494,213.

Mathematica 7 includes the function FiniteGroupCount, which will tell you the number of groups of a given order, up to 2047.
 
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I should add: In general, it's quite hard to find the number of groups of a given order. Can you prove that there are exactly 5 groups of order 8? 5 groups of order 12? It's not trivial.

Of course, for certain cases it's easy: Let p be a prime. Then there is exactly one group of order p (the cyclic one) and exactly two groups of order p2 (there are two abelian ones for sure, and it's a bit harder to show that every group of order p2 is abelian (hint: use the class equation)). It's harder to show that there are exactly five groups of order p3, but it's true.
 
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This seems such a fundamental question that more emphasis should be put on it. Is it a famous open problem as groups are so widely used.
 
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