Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#17382 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×C22⋊C4 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C22×C22⋊C4 |
| prime radical | C2×C5×#121 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C42×C5 ◁ G | |
1022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,14,16,11)(9,12)(10,13)(15,20,21,18)(17,19)(22,23,26,30,27,31,25,29)(24,28,34,37,32,35,33,36);(6,8)(7,11)(9,15)(10,16)(12,18)(13,14)(17,21)(19,20)(22,24)(23,28)(25,33)(26,34)(27,32)(29,36)(30,37)(31,35);(6,9,10,17)(7,12,13,19)(8,15,16,21)(11,18,14,20)(24,32)(28,35)(33,34)(36,37);(22,25,27,26)(23,29,31,30)(24,33,32,34)(28,36,35,37)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10370);