Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#13455 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C23×C5 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(6,7,10,12)(8,13,9,11)(14,15)(16,20)(17,21)(18,23)(19,25)(22,30)(24,31)(26,39)(27,36)(28,34)(29,35)(32,38)(33,40)(37,43)(41,45)(42,44);(6,8,10,9)(7,11,12,13)(14,16)(15,20)(17,27)(18,26)(19,29)(21,36)(22,33)(23,39)(24,34)(25,35)(28,31)(30,40)(32,41)(37,44)(38,45)(42,43);(6,9,10,8)(7,11,12,13)(15,21)(16,26)(17,30)(18,31)(20,33)(22,24)(23,37)(25,38)(27,41)(28,29)(32,43)(34,45)(35,44)(36,39)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6443);