Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#11973 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#102 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| prime radical | C23×C5 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
288 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)(6,7,9,13,10,12,18,20)(8,14,15,21,16,11,17,19)(22,23,27,32,25,34,41,31,28,33,44,49,42,30,26,43)(24,35,38,47,40,48,53,51,39,50,52,29,36,46,37,45);(6,8,10,16)(7,11,12,14)(9,17,18,15)(13,21,20,19)(22,24,28,39)(23,29,33,47)(25,36,42,40)(26,38,41,52)(27,37,44,53)(30,45,34,51)(31,48,43,46)(32,50,49,35);(7,12)(11,14)(13,20)(19,21)(22,25,28,42)(23,30,33,34)(24,36,39,40)(26,27,41,44)(29,45,47,51)(31,32,43,49)(35,48,50,46)(37,52,53,38)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4961);