Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7540 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#7541 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1415 |
| commutator | C32 | C2×C4×C5 |
| Frattini, cosocle | C2×C32 | C22×C5 |
| prime radical | C2×C5×D32 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×C8 ◁ C2×C5×C16 ◁ C2×C5×C32 ◁ G | |
94 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 69 points:
(1,2,3,4,5)(6,7,9,16,13,19,12,37)(8,20,23,47,27,14,26,38)(10,30,28,63,34,43,33,65)(11,35,29,51,36,66,32,18)(15,41,44,21,46,54,67,24)(17,49,48,22,42,57,64,31)(25,60,55,40,62,50,58,53)(39,68,45,59,52,61,69,56);(7,14)(8,21)(9,12)(10,31)(11,32)(15,42)(16,47)(17,46)(18,50)(19,20)(22,34)(23,41)(24,27)(25,61)(26,54)(28,57)(29,36)(30,39)(33,49)(35,40)(37,38)(43,52)(44,48)(45,65)(51,60)(53,66)(55,59)(56,58)(62,68)(63,69)(64,67)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 529);