Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#24473 |
| 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 | |
1314 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)(6,7,12,16,13,10,11,28)(8,17,20,14,22,31,21,19)(9,23,26,25,27,33,29,15)(18,35,36,32,37,30,24,34);(6,8)(7,14)(9,24)(10,19)(11,21)(12,20)(13,22)(15,32)(16,31)(17,28)(18,26)(23,30)(25,34)(27,36)(29,37)(33,35);(6,9)(7,15)(8,18)(10,25)(11,26)(12,29)(13,27)(14,30)(16,33)(17,34)(19,35)(20,24)(21,36)(22,37)(23,28)(31,32);(6,10,13,7)(8,19,22,14)(9,25,27,15)(11,16,12,28)(17,21,31,20)(18,35,37,30)(23,26,33,29)(24,34,36,32)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17461);