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×#34389 |
| 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 | |
1926 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)(11,15)(12,18)(17,22)(19,24)(27,31)(28,34)(29,37)(39,41);(11,15)(16,21)(17,22)(23,25)(26,27,30,31)(28,35,34,32)(29,33,37,38)(36,41,40,39);(10,11)(12,16)(13,17)(14,15)(18,21)(19,23)(20,22)(24,25)(26,27)(28,32)(29,33)(30,31)(34,35)(36,39)(37,38)(40,41);(10,12)(11,16)(13,19)(14,18)(15,21)(17,23)(20,24)(22,25)(26,28,30,34)(27,32,31,35)(29,36,37,40)(33,39,38,41);(10,13,14,20)(11,17,15,22)(12,19,18,24)(16,23,21,25)(26,29,30,37)(27,33,31,38)(28,36,34,40)(32,39,35,41)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27377);