Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19186 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#19224 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3212 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
596 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)(7,11)(9,16)(12,17)(13,18)(15,20)(19,21)(22,23)(24,31)(25,30)(26,28)(27,34)(29,32)(33,37)(35,36);(6,7,9,11,10,13,16,18)(8,12,15,17,14,19,20,21)(23,28)(29,36)(30,34)(31,37);(6,8)(7,12)(9,15)(10,14)(11,17)(13,19)(16,20)(18,21)(22,24)(23,29)(25,32)(26,33)(27,35)(28,36)(30,37)(31,34);(8,14)(12,19)(15,20)(17,21)(22,25,26,27)(23,30,28,34)(24,32,33,35)(29,37,36,31)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12212);