The 24 outer edges cannot be flipped, since the interior shape of those pieces is asymmetrical.
The full number is precisely 282 870 942 277 741 856 536 180 333 107 150 328 293 127 731 985 672 134 721 536 000 000 000 000 000 possible permutations[REF] (about 283 duodecillion on the long scale or 283 tresvigintillion on the short scale).