Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29610 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#29655 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,10,13)(8,14,16,11)(9,12,17,19)(15,20,21,18)(22,23)(24,29)(25,34)(26,31)(27,36)(28,32)(30,35)(33,37);(6,8,10,16)(7,11,13,14)(9,15,17,21)(12,18,19,20)(22,24)(23,29)(25,28)(26,33)(27,35)(30,36)(31,37)(32,34);(6,9)(7,12)(8,15)(10,17)(11,18)(13,19)(14,20)(16,21)(22,25,28,24)(23,30,32,37)(26,35,27,33)(29,36,34,31);(9,17)(12,19)(15,21)(18,20)(22,26,28,27)(23,31,32,36)(24,33,25,35)(29,37,34,30)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 22643);