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 |
| direct product | C5×#13732 |
| 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 | |
146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 77 points:
(1,2,3,4,5)(6,7,9,11)(8,12,13,10)(14,15,17,21)(16,26,28,49)(18,35,32,62)(19,38,33,63)(20,39,34,25)(22,43,40,29)(23,46,41,27)(24,48,42,36)(30,60,56,77)(31,61,57,52)(37,66,54,65)(44,72,68,76)(45,59,67,71)(47,55,64,73)(50,70,74,53)(51,69,75,58);(6,8,9,13)(7,10,11,12)(15,21)(22,40)(23,41)(24,42)(25,39)(26,49)(35,62)(38,63)(44,68)(45,67)(47,64)(50,74)(51,75)(52,61)(60,77)(65,66);(14,16,27,53,19,30,54,71,34,57,73,72,48,69,32,43,17,28,46,70,33,56,37,59,20,31,55,76,36,58,18,29)(15,22,35,51,24,44,64,61,39,67,66,77,63,74,41,49,21,40,62,75,42,68,47,52,25,45,65,60,38,50,23,26)350 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6720);