Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#11943 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #102 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| socle | C22×C5 | C2×#102 |
| prime radical | C5×#466 | C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
306 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)(22,23,26,31,27,32,41,49,28,33,42,50,40,48,25,30)(24,34,36,46,37,47,35,51,38,52,39,29,43,44,53,45);(6,8)(7,12)(9,17)(10,19)(11,18)(13,20)(14,16)(15,21)(22,24)(23,29)(25,39)(26,35)(27,43)(28,38)(30,47)(31,34)(32,51)(33,46)(36,41)(37,40)(42,53)(44,49)(45,48)(50,52);(6,9,11,10)(7,13,14,15)(8,17,18,19)(12,20,16,21)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4931);