Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#12604 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C2×D4 |
| commutator | C4 | C2×C4×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C2×C22⋊C8 |
| prime radical | C2×C5×#41 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7,9,8)(10,11)(12,21)(13,26)(14,28)(15,18)(16,20)(17,23)(19,30)(22,34)(24,38)(25,31)(27,33)(29,40)(32,39)(35,36)(37,41);(10,12,14,24,15,25,29,39,16,22,30,36,13,23,27,37)(11,17,19,32,18,21,33,35,20,31,28,41,26,34,40,38);(6,8,9,7)(12,22)(17,31)(21,34)(23,25)(24,36)(32,41)(35,38)(37,39)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5592);