Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24295 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#24295 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2408 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1302 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)(8,10)(9,22)(11,30)(12,17)(13,18)(14,20)(15,26)(16,28)(19,31)(21,29)(23,24)(25,33)(27,32)(34,37)(35,36);(6,8,11,20,13,21,12,16)(7,14,17,10,18,28,30,29)(9,23,25,19,27,34,26,35)(15,24,32,36,33,37,22,31);(6,9)(7,15)(8,19)(10,24)(11,25)(12,26)(13,27)(14,31)(16,23)(17,32)(18,33)(20,34)(21,35)(22,30)(28,36)(29,37);(6,10,11,28,13,29,12,14)(7,16,17,8,18,20,30,21)(9,24,25,36,27,37,26,31)(15,23,32,19,33,34,22,35)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17283);