Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#24287 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #535 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | #2408 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ G | |
1298 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)(8,20)(9,24)(10,28)(11,18)(12,17)(13,19)(14,22)(15,26)(16,25)(21,32)(23,37)(27,33)(29,36)(30,34)(31,35);(6,8,11,22,13,23,12,30)(7,14,17,20,19,34,18,37)(9,25,26,29,27,21,31,10)(15,32,24,36,35,16,33,28);(6,9)(7,15)(8,21)(10,22)(11,26)(12,31)(13,27)(14,32)(16,34)(17,24)(18,33)(19,35)(20,36)(23,25)(28,37)(29,30);(6,10,11,25,13,29,12,21)(7,16,17,28,19,32,18,36)(8,9,22,26,23,27,30,31)(14,33,20,15,34,24,37,35)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17275);