Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13694 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#13726 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C5 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23×C5 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ C4×C5×C8 ◁ C4×C5×C16 ◁ G | |
230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 73 points:
(1,2,3,4,5)(7,9)(10,11,16,21)(12,22,27,48)(13,28,31,60)(14,32,34,62)(15,35,33,20)(17,36,39,24)(18,40,42,23)(19,43,44,29)(25,54,56,73)(26,57,55,47)(30,61,50,59)(37,67,68,72)(38,53,63,66)(41,51,58,69)(45,65,70,49)(46,64,71,52);(10,12,23,49,14,25,50,66,33,55,69,67,43,64,31,36,16,27,40,65,34,56,30,53,15,26,51,72,29,52,13,24)(11,17,28,46,19,37,58,57,35,63,61,73,62,70,42,48,21,39,60,71,44,68,41,47,20,38,59,54,32,45,18,22);(6,7,8,9)350 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6714);