Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8674 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#8689 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #380 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C23×C5 | C22⋊Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,11,14,12,9,10,20)(8,15,18,16,19,21,17,13)(22,23)(24,30)(25,27)(26,34)(28,32)(29,33)(31,37)(35,36);(6,8,11,18,12,19,10,17)(7,13,20,21,9,16,14,15)(23,27)(28,36)(29,34)(30,37);(6,9)(7,12)(8,16)(10,20)(11,14)(13,19)(15,18)(17,21)(22,24,25,31)(23,28,27,36)(26,35,33,32)(29,37,34,30)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1677);