Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33856 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33857 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | D4×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C5×#443 ◁ G | |
646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 69 points:
(1,2,3,4,5)(6,7,13,19)(8,20,25,55)(9,26,30,51)(10,31,36,49)(11,18,12,17)(14,39,43,23)(15,37,46,29)(16,34,50,35)(21,41,27,64)(22,40,61,44)(24,62,38,47)(28,65,66,45)(32,52,56,42)(33,48,67,69)(53,59,68,58)(54,60,63,57);(6,8)(7,14)(9,27)(10,32)(11,24)(12,38)(13,25)(15,44)(16,47)(17,42)(18,52)(19,43)(20,31)(21,30)(22,28)(23,35)(26,63)(29,59)(33,60)(34,39)(36,56)(37,58)(40,46)(41,45)(48,68)(49,55)(50,62)(51,54)(53,69)(57,67)(61,66)(64,65);(6,9)(7,15)(8,21)(10,28)(11,37)(12,29)(13,30)(14,40)(16,45)(17,51)(18,26)(19,46)(20,53)(22,56)(23,57)(24,59)(25,27)(31,48)(32,61)(33,35)(34,67)(36,66)(38,58)(39,60)(41,62)(42,63)(43,44)(47,64)(49,69)(50,65)(52,54)(55,68);(6,9)(7,15)(8,22)(10,33)(11,29)(12,37)(13,30)(14,41)(16,48)(17,26)(18,51)(19,46)(20,54)(21,56)(23,58)(24,60)(25,61)(27,32)(28,35)(31,45)(34,66)(36,67)(38,57)(39,59)(40,62)(42,68)(43,64)(44,47)(49,65)(50,69)(52,53)(55,63)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26845);