Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#21000 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#121 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#386 |
| prime radical | C2×C5×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1704 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)(8,12,17,26)(9,19,20,31)(10,15)(11,14)(13,21,28,23)(16,32)(18,27,22,30)(24,29)(25,35)(33,36)(34,37);(6,8)(7,12,14,26)(9,16)(10,22)(11,17)(13,25,28,37)(15,30,29,27)(18,24)(19,36,31,32)(20,33)(21,35)(23,34);(6,9)(7,13)(8,16)(10,23)(11,20)(12,25)(14,28)(15,31)(17,33)(18,35)(19,29)(21,24)(22,34)(26,37)(27,36)(30,32);(8,17)(12,26)(16,33)(18,22)(25,37)(27,30)(32,36)(34,35)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13988);