I've been working on a problem that turns out to have a group structure. Group is of order 512. Abelian, non cyclic. Would like to find which group. I see that there are 10 or so Abelian groups of order 512. Any where I can can find a catalogue of these so I can find which one I have. Or some process to work through to that will classify the group?

GAP and Magma contain a list of "small order groups". (You need the extra package for data for order 512, 768 and between 1000 and 2000 except 1024)

Edited June 8, 200916 yr by ajb

Z512

Z256*Z2

Z128*Z4

Z128*Z2*Z2

Z64*Z8

Z64*Z4*Z2

Z64*Z2*Z2*Z2

Z32*Z16

Z32*Z8*Z2

Z32*Z4*Z4

Z32*Z4*Z2*Z2

Z32*Z2*Z2*Z2*Z2

Z16*Z16*Z2

Z16*Z8*Z4

Z16*Z8*Z2*Z2

Z16*Z4*Z4*Z2

Z16*Z4*Z2*Z2*Z2

Z16*Z2*Z2*Z2*Z2*Z2

Z8*Z8*Z8

Z8*Z8*Z4*Z2

Z8*Z8*Z2*Z2*Z2

Z8*Z4*Z4*Z4

Z8*Z4*Z4*Z2*Z2

Z8*Z4*Z2*Z2*Z2*Z2

Z8*Z2*Z2*Z2*Z2*Z2*Z2

Z4*Z4*Z4*Z4*Z2

Z4*Z4*Z4*Z2*Z2*Z2

Z4*Z4*Z2*Z2*Z2*Z2*Z2

Z4*Z4*Z2*Z2*Z2*Z2*Z2*Z2

Z2*Z2*Z2*Z2*Z2*Z2*Z2*Z2*Z2

 

Way more than 10 possibilities.

The way I suggest: Find the element of largest magnitude, and mod the group by it. Repeat until you have the trivial group. The list of orders should be the group itself. Note: You must use the largest-magnitude element, as otherwise, you could be modding out incorrectly.

Very helpful. Thank you. Highest order element is order 2.