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×#34405 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C23 | C25×C5 |
| prime radical | C2×C5×#121 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
1990 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)(26,27)(28,31)(29,34)(30,35)(32,36)(33,38)(37,39)(40,41);(11,15)(16,21)(17,22)(23,25)(27,28)(29,36)(33,35)(37,41);(10,11)(12,16)(13,17)(14,15)(18,21)(19,23)(20,22)(24,25)(26,28,31,27)(29,34,36,32)(30,35,38,33)(37,40,41,39);(10,12,14,18)(11,16,15,21)(13,19,20,24)(17,23,22,25)(26,29)(27,32)(28,34)(30,37)(31,36)(33,39)(35,40)(38,41);(10,13,14,20)(11,17,15,22)(12,19,18,24)(16,23,21,25)(26,30)(27,33)(28,35)(29,37)(31,38)(32,39)(34,40)(36,41)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27393);