Dear GAP Forum,

Gordon Royle asks:

- is the order in which the groups appear arbitrary or are
there some principles underlying it? I notice that the
abelian groups don't seem to appear as the first few groups
of each order; this is mildly surprising for lists of groups,
but maybe there is a deeper underlying principle of which I
am not aware.

Yes, there are clear principles underlying its chosen ordering.
They are described in two sources.

1. "A millenium project: classifying small groups"
by Besche, Eick and O'Brien is a survey paper
on the group construction problem. In addition
to other relevant material, it outlines
some of the organisational principles used.
 The preprint of the accepted paper is available
from my homepage
http://www.math.auckland.ac.nz/~obrien
Follow the link to Papers.

2. Some more information on the technical details of the
SmallGroups library can be found on the official homepage
of the SmallGroups project:
http://www.math.rwth-aachen.de/~Hans-Ulrich.Besche/small.html

- is this order guaranteed to remain unchanged (so that if I
refer in a paper to "the 5044th group of order 512" then this
can definitely be recovered by the future reader)

GAP and other systems provide access to the data of the library,
but the responsibility for the library contents rest with
the developers, who guarantee that its current published
ordering will not be changed.

Best wishes,
Eamonn