Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#24422 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2410 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1300 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)(6,7,12,16,11,9,13,17)(8,14,21,29,20,23,22,18)(10,15,26,30,25,24,27,31)(19,28,34,32,33,36,35,37);(6,8,11,20)(7,14,9,23)(10,19,25,33)(12,22,13,21)(15,28,24,36)(16,18,17,29)(26,35,27,34)(30,37,31,32);(6,9)(7,11)(8,18)(10,24)(12,16)(13,17)(14,22)(15,25)(19,32)(20,29)(21,23)(26,30)(27,31)(28,34)(33,37)(35,36);(6,10)(7,15)(8,19)(9,24)(11,25)(12,27)(13,26)(14,28)(16,31)(17,30)(18,32)(20,33)(21,35)(22,34)(23,36)(29,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17410);