Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#8154 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C2×C22⋊C4 |
| commutator | C23 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C22×C42×C5 ◁ G | |
602 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4,5)(6,7,9,8)(10,11)(12,16)(13,15)(14,17)(18,19)(20,23)(21,25)(22,24)(26,30)(27,29)(28,31)(32,33);(10,12)(11,14,15,16)(13,17)(18,20,21,26,22,27,28,32)(19,23,25,30,24,29,31,33);(6,8,9,7)(19,24)(23,29)(25,31)(30,33)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1142);