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) = C24 |
| direct product | C5×#12549 |
| 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×C4⋊C8 |
| prime radical | C24×C5 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C23×C5×C8 ◁ G | |
586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,14,19,15,18,28,33,16,20,29,34,13,26,27,39)(12,21,24,36,25,31,38,40,22,35,37,41,23,17,30,32);(7,9)(10,12)(11,17,20,31)(13,23)(14,24)(15,25)(16,22)(18,21,26,35)(19,32,34,40)(27,30)(28,38)(29,37)(33,36,39,41);(12,22)(17,31)(21,35)(23,25)(24,37)(30,38)(32,40)(36,41)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5537);