Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#11685 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #327 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C2×M16 | C23×C5 |
| socle | C2×C5 | C2×#327 |
| prime radical | C5×#3054 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C22×C5×M16 ◁ G | |
448 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)(8,17,20,34)(9,14)(10,23,25,37)(11,27)(12,16)(13,19)(15,22,32,26)(18,33,21,31)(24,28)(29,35)(30,36);(6,8)(7,13)(9,18)(10,24)(11,21)(12,20)(14,28)(15,31)(16,30)(17,23)(19,26)(22,36)(25,35)(27,29)(32,33)(34,37);(6,9,12,11)(7,14,16,27)(8,18,20,21)(10,22,25,26)(13,28,30,29)(15,23,32,37)(17,33,34,31)(19,24,36,35)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4673);