Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#34372 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C23 | C25×C5 |
| prime radical | C24×C5 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
1910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 41 points:
(1,2,3,4,5)(6,7,8,9)(10,11,15,13)(12,16,20,18)(14,21,22,17)(19,24,25,23)(27,31)(28,34)(29,37)(39,41);(11,13)(16,18)(17,21)(23,24)(26,27,30,31)(28,35,34,32)(29,33,37,38)(36,41,40,39);(10,12,15,20)(11,16,13,18)(14,19,22,25)(17,23,21,24)(26,27)(28,32)(29,33)(30,31)(34,35)(36,39)(37,38)(40,41);(10,13,15,11)(12,18,20,16)(14,21,22,17)(19,24,25,23)(26,28,30,34)(27,32,31,35)(29,36,37,40)(33,39,38,41);(10,14)(11,17)(12,19)(13,21)(15,22)(16,23)(18,24)(20,25)(26,29)(27,33)(28,36)(30,37)(31,38)(32,39)(34,40)(35,41)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27360);