The notation 123-456 means: stick the squares 1,2,3,4,5,6 on top of the faces ⚀,⚁,⚂,⚃,⚄,⚅ 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 ⚀ ⚁ ⚂ (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
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.
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.