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×#33860 |
| 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 | |
636 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)(6,7,14,20)(8,21,26,56)(9,27,16,42)(10,31,34,30)(11,35,40,51)(12,19,13,18)(15,43,47,24)(17,38,52,39)(22,23,28,61)(25,63,41,49)(29,59,48,33)(32,66,44,45)(36,53,68,46)(37,50,69,67)(54,55,60,58)(57,62,65,64);(6,8,14,26)(7,15,20,47)(9,28,16,22)(10,32,34,44)(11,36,40,68)(12,25,13,41)(17,49,52,63)(18,46,19,53)(21,51,56,35)(23,59,61,33)(24,38,43,39)(27,64,42,62)(29,66,48,45)(30,55,31,58)(37,60,69,54)(50,57,67,65);(6,9,14,16)(7,10,20,34)(8,22,26,28)(11,29,40,48)(12,31,13,30)(15,44,47,32)(17,33,52,59)(18,42,19,27)(21,54,56,60)(23,49,61,63)(24,57,43,65)(25,55,41,58)(35,37,51,69)(36,45,68,66)(38,50,39,67)(46,64,53,62);(6,10)(7,16)(8,23)(9,20)(11,37)(12,27)(13,42)(14,34)(15,45)(17,50)(18,31)(19,30)(21,55)(22,49)(24,62)(25,60)(26,61)(28,63)(29,51)(32,36)(33,39)(35,48)(38,59)(40,69)(41,54)(43,64)(44,68)(46,57)(47,66)(52,67)(53,65)(56,58)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26848);