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×#33859 |
| 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)(21,41,27,65)(22,40,62,44)(24,63,38,47)(28,66,60,45)(32,52,56,42)(33,48,67,69)(53,59,68,58)(54,61,64,57);(6,8,13,25)(7,14,19,43)(9,27,30,21)(10,32,36,56)(11,24,12,38)(15,44,46,40)(16,47,50,63)(17,42,18,52)(20,49,55,31)(22,60,62,28)(23,34,39,35)(26,64,51,54)(29,59,37,58)(33,61,67,57)(41,66,65,45)(48,68,69,53);(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)(32,62)(33,35)(34,67)(36,60)(38,58)(39,61)(41,63)(42,64)(43,44)(47,65)(49,69)(50,66)(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)(24,61)(25,62)(27,32)(28,35)(31,45)(34,60)(36,67)(38,57)(39,59)(40,63)(42,68)(43,65)(44,47)(49,66)(50,69)(52,53)(55,64)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26847);