Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1795 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#10767 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #379 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | C22⋊#95 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1148 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)(6,7,10,17,12,16,11,15)(8,18,21,28,22,29,20,13)(9,23,25,31,26,14,24,30)(19,33,34,37,35,32,36,27);(6,8,11,20,12,22,10,21)(7,13,15,29,16,28,17,18)(9,19,25,34,26,35,24,36)(14,27,30,33,23,37,31,32);(6,9)(7,14)(8,19)(10,24)(11,25)(12,26)(13,27)(15,30)(16,23)(17,31)(18,32)(20,34)(21,36)(22,35)(28,37)(29,33)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3755);