Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#21843 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G | |
970 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7,8,9)(10,11)(12,18)(13,16)(14,17)(15,20)(19,24)(21,23)(22,25)(26,27)(28,31)(29,32)(30,34)(33,37)(35,39)(36,38)(40,41);(10,12,14,20)(11,15,17,18)(13,21,22,19)(16,24,25,23)(26,28)(27,31)(29,36)(30,35)(32,38)(33,40)(34,39)(37,41);(10,13,14,22)(11,16,17,25)(12,19,20,21)(15,23,18,24)(27,32)(29,37)(31,38)(33,34)(36,41)(39,40);(11,17)(15,18)(16,25)(23,24)(28,35)(31,39)(36,41)(38,40)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14831);