Group of Given Order

[Matt00hew]

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?

[ajb]

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, 2009 by ajb

[uncool]

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.

[Matt00hew]

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