Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5×C8 · Schur multiplier (H2) = C4 |
| direct product | C5×#7198 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C5 | C22⋊C4 |
| commutator | C2×C4 | C4×C5×C8 |
| Frattini, cosocle | C2×C4×C8 | C22×C5 |
| socle | C22×C5 | #336 |
| prime radical | C23×C5 | #97 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 2 generators on 53 points:
(1,2,3,4,5)(6,7)(8,14,15,11)(9,13)(10,12)(16,20,21,18)(17,19)(22,23,26,31,27,32,41,49,28,33,42,50,25,30,39,47)(24,34,36,51,35,45,52,48,37,46,40,53,38,29,43,44);(6,8,9,16,10,15,17,21)(7,11,13,18,12,14,19,20)(22,24)(23,29)(25,38)(26,40)(27,35)(28,37)(30,46)(31,48)(32,34)(33,45)(36,42)(39,52)(41,43)(44,50)(47,51)(49,53)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 186);