Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C5 · Schur multiplier (H2) = C213 |
| direct product | C5×#62743 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C26×C5 |
| socle | C22×C5 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ G | |
6634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 6 generators on 37 points:
(1,2,3,4,5)(6,7)(8,14)(9,13)(10,12)(11,15)(16,20)(17,19)(18,21)(22,23)(24,27)(25,32)(26,29)(28,33)(30,36)(31,35)(34,37);(6,8)(7,11)(9,16)(10,15)(12,14)(13,18)(17,21)(19,20)(22,24)(23,27)(25,30)(26,31)(28,34)(29,35)(32,36)(33,37);(8,15)(11,14)(16,21)(18,20)(22,25,26,33)(23,28,29,32)(24,30,31,37)(27,34,35,36);(7,12)(11,14)(13,19)(18,20)(24,31)(27,35)(30,37)(34,36);(23,29)(27,35)(28,32)(34,36);(6,9,10,17)(7,13,12,19)(8,16,15,21)(11,18,14,20)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55731);