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×#34378 |
| 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 | |
2010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 33 points:
(1,2,3,4,5)(6,7,8,9)(11,14)(12,16)(18,19)(20,27)(21,24)(22,30)(23,26)(25,31)(28,32)(29,33);(11,14)(15,17)(18,20,23,21)(19,24,26,27)(22,28,31,29)(25,32,30,33);(10,11)(12,15)(13,14)(16,17)(18,21,23,20)(19,24,26,27)(22,29,31,28)(25,32,30,33);(10,12,13,16)(11,15,14,17)(18,19)(20,27)(21,24)(22,25)(23,26)(28,33)(29,32)(30,31);(18,22)(19,25)(20,28)(21,29)(23,31)(24,32)(26,30)(27,33)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27366);