Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#7545 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C5 | D32 |
| commutator | C32 | C2×C4×C5 |
| Frattini, cosocle | C2×C32 | C22×C5 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C4×C5 ◁ C2×C5×C8 ◁ C2×C5×C16 ◁ C2×C5×C32 ◁ G | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 2 generators on 73 points:
(1,2,3,4,5)(6,7,8,9)(11,17)(12,18)(13,23)(14,27)(15,30)(19,41)(20,45)(21,48)(22,36)(24,52)(25,55)(26,40)(28,57)(29,51)(31,56)(32,33)(34,42)(35,46)(37,38)(39,53)(43,70)(44,68)(47,71)(49,50)(54,73)(58,67)(59,66)(60,62)(61,72)(63,69)(64,65);(10,11,18,33,13,19,37,45,27,42,52,67,50,59,25,35,15,21,39,61,28,43,54,63,31,47,65,68,51,60,26,36,16,22,40,62,29,44,64,71,56,69,73,70,57,72,53,48,30,46,55,66,49,58,24,34,14,20,38,41,23,32,12,17)350 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 533);