linearly arranging the group tables of groups of order 128 I'm planning to make a video that shows color coded group tables for all 2328 groups of order 128 -- at 128$\times$128 pixels at 24 frames a second I think I get 97 seconds of video. Is there some logical order that the all these groups can be placed in that would make the most visual sense? 
It's done! Here: http://www.youtube.com/watch?v=25Qzu-KuVs4
 A: I would be inclined to follow the ordering in the GAP/Magma libraries, which sorts first by number of generators. So you get the cyclic group (tall and thin) first and the elementary abelian group (short and wide) last. Of course the huge majority of the groups have 3,4 or 5 generators.
Groups with the same generator number are sorted in ascending order of exponent $p$-class (which is similar to but not quite the same as nilpotency class).  But again you get huge numbers of group with, for example, 4 generators and class 2, which are all very similar to each other. So perhaps the video should go slowly at the beginning and end, when group properties are changing more rapidly, and speed up during the repetitive middle sections!
A: Something like this?

(This is a movie of the Cayley tables of quandles of order five.  I could not upload the order seven file; I guess it was too big.)  I generated these movies in row-major lexicographic order.  You could do the same or, alternatively, you could do those groups in the order they occur in the GAP/Magma libraries of small groups.
But, for groups of order $128$ there might be a partial order that would be visually interesting: sort by the size of the centre.  This would put the abelian ones together.  For groups with the same size centres, you could recursively sort according to the same criterion applied to their central factor groups.  Since your groups are $2$-groups, this will eventually terminate, and you more-or-less work your way along the upper central series.  (When you get down to abelian groups, you could break ties by looking at the vectors of sizes of cyclic factors.)  Alternatively, you could go down instead of up.
EDIT: Out of curiosity, I tried doing the latter (going down the lower central series) for the groups of orders 16 and 32.  (Let's see if I can upload the more interesting order 32 movie ... nope.  Let's try the order 16 ... okay.)  It is perhaps not as exciting as I had first imagined, though you might have better luck with the upper central series (or lower $2$-central series).  I didn't try playing with the colours or frame rate, which might be more usefully chosen to reflect the structure.

