Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#20919 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×C4×D4 |
| prime radical | C22×C5×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C5×M16 ◁ G | |
1682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(7,9)(10,11)(12,22,24,32)(13,27,30,35)(14,20)(15,21)(16,19)(17,25,33,26)(18,29,34,28)(23,37)(31,38)(36,39)(40,41);(6,7,8,9)(10,12,14,26,16,24,15,25)(11,17,20,32,19,33,21,22)(13,23,29,39,30,38,28,40)(18,31,27,41,34,37,35,36);(10,13)(11,18)(12,23)(14,29)(15,28)(16,30)(17,31)(19,34)(20,27)(21,35)(22,36)(24,38)(25,40)(26,39)(32,41)(33,37);(11,19)(12,24)(18,34)(20,21)(23,38)(25,26)(27,35)(39,40)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13907);