perhaps i should give the figures i obtained when getting my upper
bound of 39 face / 56 quarter turns in case i also have an error.
recall the method: we work in three stages:

1. from  <U, F, R, L, B, D>  to  <U, R, F>
2. from  <U, R, F>           to  <U, R2, F2>
3. from  <U, R2, F2>         to  START

where we're only allowed to use moves that keep us within the
given subgroup. the number of positions that must be considered
in each stage are 253440, 15676416, and 10886400 respectively.

stage 1:
using face turns:          using quarter turns:
distance  number           distance     number
  0            1               0             1
  1            9               1             6
  2           90               2            39
  3          852               3           276
  4         7169               4          1899
  5        44182               5         11245
  6       131636               6         49412
  7        68940               7        117221
  8          561               8         70767
                               9          2574
stage 2:
using face turns:          using quarter turns:
distance  number           distance     number
  0            1               0             1
  1            2               1             2
  2           12               2             8
  3           72               3            36
  4          420               4           158
  5         2410               5           694
  6        13752               6          2980
  7        75796               7         12744
  8       390421               8         53646
  9      1735771               9        216354
 10      5351383              10        799868
 11      6696700              11       2477802
 12      1399195              12       5310848
 13        10481              13       5419046
                              14       1356020
                              15         26192
                              16            17
stage 3:
using face turns:          using quarter turns:
distance  number           distance     number
  0            1               0             1
  1            5               1             2
  2           14               2             3
  3           44               3             8
  4          128               4            14
  5          392               5            24
  6         1157               6            52
  7         3458               7            96
  8        10057               8           176
  9        29286               9           352
 10        82814              10           664
 11       228621              11          1248
 12       591704              12          2409
 13      1362497              13          4516
 14      2545752              14          8519
 15      3272940              15         16100
 16      2260555              16         30171
 17       484818              17         56140
 18        12133              18        102981
 19           24              19        186728
                              20        331234
                              21        563985
                              22        912719
                              23       1365051
                              24       1812011
                              25       2044832
                              26       1783956
                              27       1105488
                              28        450322
                              29         97881
                              30          7958
                              31           745
                              32            10
                              33             4

it would be nice if someone could confirm these figures. i have done
some testing, but not extensively.

mike