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×#11946 |
| 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 | |
302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5)(6,7,10,15,11,14,9,13)(8,16,17,21,18,12,19,20)(22,23,25,31,39,45,52,47,26,29,40,30,27,32,41,49)(24,33,35,50,42,48,38,43,36,51,34,44,37,46,53,28);(6,8)(7,12)(9,17)(10,19)(11,18)(13,20)(14,16)(15,21)(22,24)(23,28)(25,38)(26,36)(27,42)(29,43)(30,46)(31,48)(32,50)(33,47)(34,41)(35,52)(37,39)(40,53)(44,45)(49,51);(6,9,11,10)(7,13,14,15)(8,17,18,19)(12,20,16,21)(23,29)(28,43)(30,31)(32,45)(33,51)(44,50)(46,48)(47,49)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4934);