Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18712 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18740 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,14)(9,17)(10,12)(11,15)(13,19)(16,21)(18,20)(22,23)(24,32)(25,30)(26,29)(27,31)(28,34)(33,37)(35,36);(6,8,10,16)(7,11,12,20)(9,18,19,15)(13,21,17,14)(22,24,27,35)(23,28,31,37)(25,33,26,34)(29,36,30,32);(6,7)(8,11)(9,13)(10,12)(14,15)(16,20)(17,19)(18,21)(23,29)(25,26)(28,36)(30,31)(32,37)(33,34);(7,12)(11,20)(13,17)(14,21)(22,25,27,26)(23,29,31,30)(24,33,35,34)(28,36,37,32)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11728);