Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5×C8 · Schur multiplier (H2) = C29 |
| direct product | C5×#62624 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C16 | C24 |
| commutator | C2 | C24×C5×C8 |
| Frattini, cosocle | C8 | C25×C5 |
| socle | C2×C5 | C24×C8 |
| prime radical | C5×#585 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C16 ◁ G | |
2038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 69 points:
(1,2,3,4,5)(6,7,10,15,11,16,25,35,12,17,26,36,27,37,47,57)(8,13,20,30,21,31,43,53,19,29,41,51,42,52,61,66)(9,14,23,33,24,34,46,56,22,32,44,54,45,55,62,67)(18,28,39,49,40,50,60,65,38,48,58,63,59,64,68,69);(6,8)(7,13)(9,18)(10,20)(11,21)(12,19)(14,28)(15,30)(16,31)(17,29)(22,38)(23,39)(24,40)(25,43)(26,41)(27,42)(32,48)(33,49)(34,50)(35,53)(36,51)(37,52)(44,58)(45,59)(46,60)(47,61)(54,63)(55,64)(56,65)(57,66)(62,68)(67,69);(6,9)(7,14)(8,18)(10,23)(11,24)(12,22)(13,28)(15,33)(16,34)(17,32)(19,38)(20,39)(21,40)(25,46)(26,44)(27,45)(29,48)(30,49)(31,50)(35,56)(36,54)(37,55)(41,58)(42,59)(43,60)(47,62)(51,63)(52,64)(53,65)(57,67)(61,68)(66,69);(9,22)(14,32)(18,38)(23,44)(24,45)(28,48)(33,54)(34,55)(39,58)(40,59)(46,62)(49,63)(50,64)(56,67)(60,68)(65,69);(8,19)(13,29)(18,38)(20,41)(21,42)(28,48)(30,51)(31,52)(39,58)(40,59)(43,61)(49,63)(50,64)(53,66)(60,68)(65,69)680 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55612);