Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#11110 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #390 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1439 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
852 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)(6,7,11,15,12,17,10,16)(8,18,21,28,22,32,20,13)(9,23,24,14,26,31,25,30)(19,27,34,37,35,29,36,33);(6,8,10,20,12,22,11,21)(7,13,15,18,17,28,16,32)(9,19,24,34,26,35,25,36)(14,29,23,37,30,27,31,33);(6,9)(7,14)(8,19)(10,24)(11,25)(12,26)(13,27)(15,23)(16,31)(17,30)(18,33)(20,34)(21,36)(22,35)(28,29)(32,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4098);