Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9056 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | #386 |
| prime radical | C23×C5 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
432 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)(7,11,13,16)(8,14,15,17)(22,23)(24,32,35,36)(25,29,28,34)(26,30)(27,31)(33,37);(6,7,9,16,10,13,18,11)(8,12,17,20,15,21,14,19)(22,24,33,34,27,35,26,29)(23,28,37,32,31,25,30,36);(6,8)(7,12)(9,17)(10,15)(11,19)(13,21)(14,18)(16,20)(22,25,27,28)(23,29,31,34)(24,30,35,37)(26,32,33,36)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2044);