Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11704 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#11945 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#102 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| prime radical | C5×#466 | C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,10,15,11,14,9,13)(8,16,17,21,18,12,19,20)(22,23,26,30,35,28,37,33,27,31,36,29,25,24,32,34);(6,8)(7,12)(9,17)(10,19)(11,18)(13,20)(14,16)(15,21)(22,24)(23,25)(26,30)(27,28)(29,36)(31,35)(32,33)(34,37);(6,9,11,10)(7,13,14,15)(8,17,18,19)(12,20,16,21)(22,25,27,35)(23,28,31,24)(26,32,36,37)(29,34,30,33)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4933);