Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33657 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33657 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×#481 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#3058 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
566 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)(8,16)(9,19)(10,22)(12,25)(14,29)(15,28)(17,33)(18,21)(20,35)(23,36)(24,34)(26,32)(27,30)(31,37);(6,7)(8,12)(9,20)(10,15)(11,13)(14,23)(16,32)(17,26)(18,27)(19,35)(21,37)(22,28)(24,31)(25,33)(29,36)(30,34);(6,8,10,24,11,17,22,18)(7,12,15,31,13,26,28,27)(9,21,19,33,23,34,36,16)(14,30,29,32,20,37,35,25);(7,13)(12,26)(14,20)(15,28)(25,32)(27,31)(29,35)(30,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26645);