Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9996 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#10003 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C23×C5 | #120 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
500 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)(6,7,11,14)(8,15,17,20)(9,13,10,18)(12,19,21,16)(22,23)(24,30)(25,33)(26,29)(27,32)(28,35)(31,37)(34,36);(6,8,11,17)(7,12,14,21)(9,16,10,19)(13,20,18,15)(27,36)(30,37);(6,9,11,10)(7,13,14,18)(8,16,17,19)(12,20,21,15)(22,24)(23,27)(25,34)(26,31)(28,37)(29,36)(30,33)(32,35)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2991);