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) = C2×C4 |
| direct product | C5×#12665 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C2×D4 |
| commutator | C4 | C2×C4×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C8×D4 |
| prime radical | C23×C5 | #131 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C22×C5×C16 ◁ G | |
326 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)(10,11,15,19,14,18,27,13,16,20,32,28,30,37,41,29)(12,21,24,22,25,17,33,34,26,35,31,40,23,36,39,38);(6,7,8,9)(10,12)(11,17)(13,22)(14,25)(15,31)(16,26)(18,35)(19,38)(20,36)(21,37)(23,30)(24,32)(27,39)(28,34)(29,40)(33,41);(10,13,27,37,30,19,15,20,16,29,41,18,14,28,32,11)(12,22,39,21,23,38,31,36,26,40,33,35,25,34,24,17)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5653);