Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13442 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#13442 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2054 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#509 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
398 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)(7,14)(8,20)(9,25)(10,11)(12,26)(16,29)(17,19)(18,27)(21,30)(22,36)(23,28)(24,34)(31,35)(33,37);(6,7)(8,21)(9,16)(10,17)(11,27)(12,29)(13,19)(14,25)(15,22)(18,26)(20,31)(23,37)(24,33)(28,32)(30,36)(34,35);(6,8,9,22,26,36,10,20,13,24,11,28,12,23,25,34)(7,15,14,21,29,35,27,37,19,32,17,33,18,31,16,30)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6430);