Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33808 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C23×C5 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ C5×#405 ◁ G | |
442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 53 points:
(1,2,3,4,5)(6,7)(8,14)(9,12)(10,13)(11,16)(15,20)(17,19)(18,21)(22,23,27,32)(24,28,33,41)(25,29,38,45)(26,39,37,31)(30,46,44,36)(34,40,47,51)(35,50,48,43)(42,53,52,49);(6,8,10,16)(7,11,13,14)(9,17,18,15)(12,20,21,19)(22,24)(23,28)(25,34)(26,35)(27,33)(29,40)(30,42)(31,43)(32,41)(36,49)(37,48)(38,47)(39,50)(44,52)(45,51)(46,53);(6,9,10,18)(7,12,13,21)(8,15,16,17)(11,19,14,20)(23,29)(25,36)(26,37)(28,40)(30,31)(32,45)(34,49)(35,48)(38,46)(39,44)(41,51)(42,43)(47,53)(50,52);(7,13)(11,14)(12,21)(19,20)(24,33)(28,41)(34,47)(35,48)(40,51)(42,52)(43,50)(49,53)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26796);