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×#34490 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C25 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C23 | C25×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
1840 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 5 generators on 41 points:
(1,2,3,4,5)(6,7,9,8)(10,11,14,15)(12,18,19,16)(13,21,22,17)(20,23,25,24)(26,27,32,30)(28,35,29,33)(31,38,39,34)(36,40,37,41);(11,15)(12,19)(17,21)(20,25)(26,28,32,29)(27,33,30,35)(31,36,39,37)(34,40,38,41);(10,11)(12,16)(13,17)(14,15)(18,19)(20,23)(21,22)(24,25)(26,29,32,28)(27,33,30,35)(31,37,39,36)(34,40,38,41);(6,8,9,7)(10,12,14,19)(11,16,15,18)(13,20,22,25)(17,23,21,24)(26,30,32,27)(28,33,29,35)(31,38,39,34)(36,40,37,41);(10,13,14,22)(11,17,15,21)(12,20,19,25)(16,23,18,24)(26,31,32,39)(27,34,30,38)(28,36,29,37)(33,40,35,41)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27478);