Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C27×C4 |
| direct product | C5×#35320 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| prime radical | C24×C5 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
2230 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)(7,9)(12,16)(14,17)(18,19,23,26)(20,21,24,29)(22,30,31,25)(27,33,32,28);(6,7,8,9)(10,11,13,15)(12,14,16,17)(18,20)(19,21)(22,27)(23,24)(25,28)(26,29)(30,33)(31,32);(10,12,13,16)(11,14,15,17)(18,21)(19,24)(20,26)(22,28)(23,29)(25,32)(27,30)(31,33);(18,21)(19,24)(20,26)(22,28)(23,29)(25,32)(27,30)(31,33);(18,22,23,31)(19,25,26,30)(20,27,24,32)(21,28,29,33)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 28308);