Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#22690 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
846 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)(6,7,8,9)(10,11)(12,24)(13,28)(14,20)(15,22)(16,21)(17,23)(18,26)(19,29)(25,39)(27,37)(30,38)(31,41)(32,35)(33,36)(34,40);(10,12,17,27)(11,18,23,36)(13,14,30,32)(15,33,16,26)(19,35,39,20)(21,37,22,24)(25,34,29,31)(28,41,38,40);(10,13,16,29,17,30,15,25)(11,19,22,28,23,39,21,38)(12,14,26,31,27,32,33,34)(18,35,24,41,36,20,37,40);(10,14)(11,20)(12,25)(13,26)(15,34)(16,31)(17,32)(18,28)(19,37)(21,41)(22,40)(23,35)(24,39)(27,29)(30,33)(36,38)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15678);