MacMahon's coloured cubes

The notation 123-456 means: stick the squares 1,2,3,4,5,6
on top of the faces 1 2 3 4 5 6 of a standard die.
All four versions on a single line describe the same coloured cube.
Between them, they list the eight corners of the cube in the same sense
as 1 2 3 (on my die, that sense is clockwise).

Note that 123, 231, 312 represent the same corner, but
132, 213, 321 represent a corner with the same colours the other way round.

 1: 123-456 135-246 154-326 142-536 
 2: 123-465 136-245 164-325 142-635 
 3: 123-546 134-256 145-326 152-436 
 4: 123-564 136-254 165-324 152-634 
 5: 123-645 134-265 146-325 162-435 
 6: 123-654 135-264 156-324 162-534 
 7: 124-356 145-236 153-426 132-546 
 8: 124-365 146-235 163-425 132-645 
 9: 124-536 143-256 135-426 152-346 
10: 124-563 146-253 165-423 152-643 
11: 124-635 143-265 136-425 162-345 
12: 124-653 145-263 156-423 162-543 
13: 125-346 154-236 143-526 132-456 
14: 125-364 156-234 163-524 132-654 
15: 125-436 153-246 134-526 142-356 
16: 125-463 156-243 164-523 142-653 
17: 125-634 153-264 136-524 162-354 
18: 125-643 154-263 146-523 162-453 
19: 126-345 164-235 143-625 132-465 
20: 126-354 165-234 153-624 132-564 
21: 126-435 163-245 134-625 142-365 
22: 126-453 165-243 154-623 142-563 
23: 126-534 163-254 135-624 152-364 
24: 126-543 164-253 145-623 152-463 
25: 134-562 146-352 165-432 153-642 
26: 134-652 145-362 156-432 163-542 
27: 135-462 156-342 164-532 143-652 
28: 135-642 154-362 146-532 163-452 
29: 136-452 165-342 154-632 143-562 
30: 136-542 164-352 145-632 153-462 

Sobczyk's classification

Find a group of five cubes that between them contain all 40 possible corners.
(Hint: don't use cubes that are mirror images of each other.)
From the cubes that remain, find another such group. And again, and again, and again. 
The last five cubes will also form such a group.

Conway's notation

Label the six groups A,B,C,D,E,F. Form six other groups from the mirror images
of the original six groups and label them a,b,c,d,e,f. Each cube belongs to
exactly two groups. Those two letters, say Db, are its name in Conway's notation.

G4G-COM 2011 lecture by Dirk Laurie at AIMS South Africa.