Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9946 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9958 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #392 |
| commutator, Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C2×C5×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
594 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)(6,7,11,15)(8,16,18,20)(9,12,10,14)(13,19,21,17)(22,23)(24,27)(25,28)(26,29)(30,33)(31,34)(32,36)(35,37);(7,12)(9,10)(13,16)(14,15)(17,19)(20,21)(22,24,26,31)(23,27,29,34)(25,30,32,37)(28,35,36,33);(6,8)(7,13)(9,17)(10,19)(11,18)(12,20)(14,16)(15,21)(22,25)(23,28)(24,30)(26,32)(27,33)(29,36)(31,37)(34,35)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2946);