Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#20947 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#120 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#386 |
| prime radical | C5×#3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1708 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,16,19,30)(9,20,22,13)(10,15)(11,14)(12,18,27,23)(17,32)(21,31,24,28)(25,29)(26,33)(34,37)(35,36);(6,8,10,23,11,19,25,18)(7,12,15,30,14,27,29,16)(9,17,24,33,22,34,21,35)(13,26,31,37,20,36,28,32);(6,9)(7,13)(8,17)(10,24)(11,22)(12,26)(14,20)(15,31)(16,32)(18,35)(19,34)(21,25)(23,33)(27,36)(28,29)(30,37);(7,14)(12,27)(13,20)(15,29)(16,30)(26,36)(28,31)(32,37)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13935);