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×#33807 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#3176 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ C5×#405 ◁ G | |
466 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)(24,28)(25,29)(26,39)(27,32)(30,36)(31,37)(33,41)(34,40)(35,49)(38,45)(42,48)(43,47)(44,50)(46,51)(52,53);(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)(36,48)(37,47)(38,46)(39,49)(44,52)(45,51)(50,53);(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)(34,48)(35,47)(38,50)(39,44)(41,51)(42,43)(46,53)(49,52);(7,13)(11,14)(12,21)(19,20)(24,33)(28,41)(34,46)(35,47)(40,51)(42,52)(43,49)(48,53)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26795);