Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9034 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | #386 |
| prime radical | C24×C5 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,10,14,11,15,9,13)(8,16,17,12,19,20,18,21)(22,23)(24,32,34,36)(25,35,28,29)(26,30)(27,31)(33,37);(6,8)(7,12)(9,17)(10,18)(11,19)(13,20)(14,16)(15,21)(22,24,26,35,27,34,37,29)(23,28,30,36,31,25,33,32);(6,9,11,10)(7,13,15,14)(8,17,19,18)(12,20,21,16)(22,25,27,28)(23,29,31,35)(24,33,34,30)(26,32,37,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2022);