Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#26846 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C4×C5×C4⋊C4 ◁ G | |
800 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)(10,11)(12,17)(13,15)(14,16)(18,19,25,30)(20,27,31,33)(21,26,22,32)(23,29,24,28);(6,7)(8,9)(10,12,14,13)(11,15,16,17)(18,20,24,22,25,31,23,21)(19,26,29,33,30,32,28,27);(7,9)(10,13,14,12)(11,15,16,17)(18,21,23,31,25,22,24,20)(19,27,28,32,30,33,29,26);(11,16)(15,17)(18,22,24,31,25,21,23,20)(19,27,29,26,30,33,28,32)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19834);