Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#13370 |
| 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 | |
358 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)(6,7,9,11)(8,12,13,10)(14,15)(16,25)(17,31)(18,23)(19,36)(20,24)(21,27)(22,34)(26,41)(28,38)(29,45)(30,43)(32,40)(33,39)(35,42)(37,44);(6,8,9,13)(7,10,11,12)(14,15)(16,26)(17,22)(18,36)(19,23)(20,24)(21,28)(25,41)(27,38)(29,37)(30,40)(31,34)(32,43)(33,42)(35,39)(44,45);(14,16,20,30)(15,21,24,37)(17,32,35,41)(18,29,19,28)(22,38,39,45)(23,40,36,26)(25,42,43,31)(27,33,44,34)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6358);