Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#26137 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#2414 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C4×C5×D4 ◁ G | |
894 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)(6,7)(8,9)(10,11)(12,15)(13,17)(14,16)(18,19,23,22)(20,24,27,30)(21,25,29,31)(26,32,33,28);(7,9)(10,12,14,13)(11,15,16,17)(18,20)(19,24)(21,28)(22,30)(23,27)(25,26)(29,32)(31,33);(10,13,14,12)(11,15,16,17)(18,21,23,29)(19,25,22,31)(20,26,27,33)(24,32,30,28);(7,9)(18,19,23,22)(20,24,27,30)(21,25,29,31)(26,32,33,28)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19125);