Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33638 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33638 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×#476 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(7,13)(8,18,20,36)(9,11)(10,22)(14,28,29,32)(15,17)(19,23,25,35)(21,24)(26,30,34,37)(27,33);(6,7,9,13,12,15,11,17)(8,14,19,26,20,29,25,34)(10,16,21,27,24,31,22,33)(18,30,35,28,36,37,23,32);(6,8)(7,14)(9,19)(10,23)(11,25)(12,20)(13,26)(15,29)(16,32)(17,34)(18,21)(22,36)(24,35)(27,30)(28,31)(33,37);(7,15)(13,17)(14,29)(16,31)(26,34)(27,33)(28,32)(30,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26626);