Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13374 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#13374 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2023 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7)(8,16,18,33)(9,21)(10,25)(11,14)(12,19,17,35)(13,23)(15,27)(20,29,26,34)(22,31,37,28)(24,32)(30,36);(6,7)(8,17,18,12)(9,13)(10,15)(11,14)(16,19,33,35)(20,31,26,28)(21,23)(22,34,37,29)(24,30)(25,27)(32,36);(6,8)(7,12)(9,22)(10,26)(11,18)(13,29)(14,17)(15,31)(16,32)(19,36)(20,27)(21,33)(23,35)(24,37)(25,28)(30,34)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6362);