Finite Group DatabasebetaFinite Group Databasebeta| order | 25856 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C101 · Schur multiplier (H2) = C25×C4 |
| direct product | C101×#23540 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C101 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C101 |
| prime radical | C23×D8×C101 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C101 ◁ C2×#117×C101 ◁ G | |
1682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 6464 |
A minimal faithful permutation representation — 4 generators on 125 points:
(1,2)(3,4)(9,10)(11,19)(12,15)(13,17)(14,18)(16,20)(21,24)(22,23)(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);(5,6)(7,8)(10,15)(11,20)(12,13)(16,23)(17,18)(21,22);(2,4)(6,8)(9,11,12,22,14,21,13,20)(10,16,17,24,18,23,15,19);(11,21)(16,23)(19,24)(20,22)5252 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(25856, 16528);