Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C2×C42 |
| direct product | C5×#16489 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C5 | C24 |
| commutator, Frattini, cosocle | C24 | C24×C5 |
| prime radical | C25×C5 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ G | |
1246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3,4,5)(7,9)(10,11)(12,16)(13,14)(15,17)(18,19,21,23)(20,22,24,25)(28,32)(31,33);(11,14)(15,16)(20,24)(22,25)(26,27,30,29)(28,31,32,33);(10,11)(12,16)(13,14)(15,17)(18,20)(19,22)(21,24)(23,25)(26,28,30,32)(27,31,29,33);(6,7)(8,9)(10,12,13,17)(11,15,14,16)(18,19)(20,22)(21,23)(24,25)(26,29,30,27)(28,33,32,31)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 9477);