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×#10766 |
| 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 | |
1180 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)(8,17)(9,21,23,14)(10,25)(11,15)(12,16)(13,19)(18,32,34,31)(20,29)(22,30,24,36)(26,28)(27,33,35,37);(6,8,11,19,12,20,10,26)(7,13,15,29,16,28,25,17)(9,18,24,33,23,34,22,37)(14,27,30,32,21,35,36,31);(6,9)(7,14)(8,18)(10,22)(11,24)(12,23)(13,27)(15,30)(16,21)(17,31)(19,33)(20,34)(25,36)(26,37)(28,35)(29,32)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3754);