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×#12656 |
| 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 | C23×C5 | C2×M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
358 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)(6,7,9,8)(10,11)(12,24,27,37)(13,29)(14,20)(15,21)(16,22)(17,23)(18,26,34,32)(19,31)(25,35,39,38)(28,33,40,41)(30,36);(10,12,15,28,16,26,31,39,17,27,30,40,14,32,13,25)(11,18,21,35,22,24,19,33,23,34,36,38,20,37,29,41);(6,8,9,7)(10,13,14,30,17,31,16,15)(11,19,20,21,23,29,22,36)(12,25,32,40,27,39,26,28)(18,33,37,35,34,41,24,38)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5644);