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×#19215 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C23×C5 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
582 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)(9,11)(10,18)(12,16)(13,17)(15,21)(19,20)(22,23,27,30)(24,31,29,26)(25,32,33,37)(28,34,36,35);(7,11)(12,20)(13,18)(14,21)(22,24,26,23,27,29,31,30)(25,32,34,28,33,37,35,36);(6,8,9,15)(7,12,11,20)(10,19,17,16)(13,21,18,14)(22,25,27,33)(23,28,30,36)(24,32,29,37)(26,34,31,35);(25,33)(28,36)(32,37)(34,35)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12203);