Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33618 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33618 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×#466 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
656 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)(7,13)(8,19)(9,14)(10,11)(15,30)(16,26)(17,28)(18,27)(20,34)(21,24)(22,23)(25,32)(31,37)(33,36);(6,7,14,28,11,18,23,26,12,16,22,27,10,17,9,13)(8,15,29,30,19,32,34,36,21,31,35,37,24,33,20,25);(6,8)(7,15)(9,20)(10,24)(11,19)(12,21)(13,25)(14,29)(16,31)(17,33)(18,32)(22,35)(23,34)(26,36)(27,37)(28,30);(7,16)(13,26)(15,31)(17,18)(25,36)(27,28)(30,37)(32,33)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26606);