Finite Group DatabasebetaFinite Group Databasebeta| order | 25856 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C101 · Schur multiplier (H2) = C23×C4 |
| direct product | C101×#19749 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C101 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C101 |
| prime radical | C101×#3151 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C101 ◁ C2×C42×C101 ◁ G | |
822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 6464 |
A minimal faithful permutation representation — 4 generators on 125 points:
(2,5)(3,7)(6,8)(9,10)(11,17)(12,20)(13,15)(14,19)(16,21)(18,23)(22,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(9,11,12,19,13,18,21,24)(10,14,16,17,15,22,20,23);(1,2,3,5,4,6,7,8)(10,15)(14,22)(16,20)(17,23);(2,6)(5,8)(11,18)(14,22)(17,23)(19,24)4040 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(25856, 12737);