Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33656 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #481 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C2×C5 | C2×#481 |
| prime radical | C5×#3054 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C22×C5×M16 ◁ G | |
556 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)(6,7)(8,17)(9,14)(10,22)(11,26)(12,16)(13,20)(15,25)(18,32)(19,33)(21,31)(23,37)(24,30)(27,34)(28,35)(29,36);(10,23)(15,31)(17,33)(20,36)(21,25)(22,37)(24,35)(32,34);(6,8,11,27,12,19,9,18)(7,13,14,28,16,29,26,30)(10,24,25,36,23,35,21,20)(15,32,22,33,31,34,37,17);(6,9,12,11)(7,14,16,26)(8,18,19,27)(10,21,23,25)(13,28,29,30)(15,22,31,37)(17,32,33,34)(20,35,36,24)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26644);