Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#20787 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #117 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | #2283 |
| prime radical | C2×C5×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C5×#405 ◁ G | |
616 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)(8,17,20,13)(9,14)(10,15)(11,16)(12,19,25,33)(18,30,32,26)(21,27)(22,28)(23,29)(24,31,35,37)(34,36);(6,8)(7,12,16,25)(9,18)(10,19)(11,20)(13,15,17,29)(14,24,28,35)(21,31)(22,32)(23,33)(26,27,30,36)(34,37);(6,8)(7,13)(9,18)(10,19)(11,20)(12,15)(14,26)(16,17)(21,31)(22,32)(23,33)(24,27)(25,29)(28,30)(34,37)(35,36);(6,9,10,21,11,22,23,34)(7,14,15,27,16,28,29,36)(8,18,19,31,20,32,33,37)(12,24,17,30,25,35,13,26)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13775);