Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#23995 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #535 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | #2404 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ G | |
1164 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)(8,12)(9,13)(10,15)(11,16)(14,18)(17,28)(19,26)(20,37)(21,29)(22,30)(23,32)(24,33)(25,34)(27,36)(31,35);(6,8)(7,12)(9,20,21,36)(10,19)(11,18)(13,27,29,37)(14,16)(15,26)(17,22,34,23)(24,35)(25,30,28,32)(31,33);(6,9)(7,13)(8,17)(10,23)(11,21)(12,25)(14,28)(15,32)(16,29)(18,34)(19,36)(20,35)(22,24)(26,37)(27,31)(30,33);(6,8)(7,14)(9,17)(10,19)(11,18)(12,16)(13,28)(15,31)(20,22)(21,34)(23,36)(24,35)(25,29)(26,33)(27,32)(30,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16983);