Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#29837 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C5×#376 ◁ G | |
712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3,4,5)(6,7,8,9)(10,11,13,14)(12,15,16,17)(18,19)(20,23)(21,27)(22,28)(24,26)(25,31)(29,32)(30,33);(11,14)(15,17)(18,20,23,19)(21,29,31,33)(22,28,24,26)(25,32,27,30);(7,9)(12,16)(15,17)(18,21,23,31)(19,25,20,27)(22,32,24,30)(26,29,28,33);(10,12,13,16)(11,15,14,17)(18,22,23,24)(19,26,20,28)(21,30,31,32)(25,33,27,29)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 22825);