Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10500 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#10506 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C23×C5 | #121 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,10,14)(8,15,17,12)(9,11,18,13)(16,20,21,19)(22,23)(24,27)(25,28)(26,30)(29,34)(31,35)(32,37)(33,36);(7,11)(8,16)(9,18)(12,15)(13,14)(17,21)(22,24,26,32)(23,27,30,37)(25,33,34,31)(28,36,29,35);(6,8,9,16,10,17,18,21)(7,12,13,20,14,15,11,19)(23,28)(25,34)(27,35)(29,30)(31,33)(36,37)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3494);