Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33687 |
| 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 | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 77 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,18,23)(16,26,28,21)(17,31,32,67)(19,37,35,66)(20,40,36,25)(22,45,46,33)(24,49,48,39)(27,55,56,65)(29,61,59,70)(30,44,60,54)(34,68,41,50)(38,51,69,47)(42,57,52,74)(43,63,53,75)(58,72,64,76)(62,71,77,73);(7,10)(11,13)(14,16)(15,21)(17,27)(18,28)(19,29)(20,30)(22,42)(23,26)(24,43)(25,44)(31,65)(32,56)(33,57)(34,58)(35,59)(36,60)(37,70)(38,62)(39,63)(40,54)(41,64)(45,74)(46,52)(47,71)(48,53)(49,75)(50,72)(51,73)(55,67)(61,66)(68,76)(69,77);(14,17,18,32)(15,22,23,46)(16,27,28,56)(19,38,35,69)(20,41,36,34)(21,42,26,52)(24,50,48,68)(25,51,40,47)(29,62,59,77)(30,64,60,58)(31,66,67,37)(33,49,45,39)(43,72,53,76)(44,73,54,71)(55,70,65,61)(57,75,74,63);(6,8,9,12)(7,11,10,13)(16,28)(21,26)(27,56)(29,59)(30,60)(42,52)(43,53)(44,54)(55,65)(57,74)(58,64)(61,70)(62,77)(63,75)(71,73)(72,76)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26675);