Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#19004 |
| 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 | |
896 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)(7,12)(8,16)(9,10)(14,15)(17,18)(19,20)(23,28)(25,26)(29,36)(30,33)(31,34)(35,37);(6,7)(8,13)(9,14)(10,12)(11,15)(16,20)(17,21)(18,19)(22,23,27,30)(24,29,32,37)(25,28,26,33)(31,35,34,36);(6,8)(7,13)(9,16)(10,18)(11,17)(12,19)(14,20)(15,21)(22,24)(23,29)(25,31)(26,34)(27,32)(28,35)(30,37)(33,36);(6,9,11,10)(7,14,15,12)(8,16,17,18)(13,20,21,19)(22,25,27,26)(23,28,30,33)(24,31,32,34)(29,35,37,36)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11992);