Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12833 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12833 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1807 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
358 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)(8,19,23,37)(9,24,26,34)(10,20)(11,16)(12,18)(13,21,27,14)(15,28)(17,30)(22,36,31,25)(29,32)(33,35);(6,8,10,21,11,22,28,34,12,23,29,14,30,31,35,24)(7,13,15,19,17,9,20,25,18,27,33,37,16,26,32,36);(6,9)(7,14)(8,20)(10,25)(11,27)(12,26)(13,30)(15,31)(16,34)(17,24)(18,21)(19,35)(22,33)(23,32)(28,37)(29,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5821);