Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9369 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9372 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C5×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
390 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)(6,7,12,15,10,9,11,14)(8,16,18,20,19,13,17,21)(22,23,27,31,28,32,26,42)(24,33,37,44,36,48,35,29)(25,38,40,46,41,30,39,45)(34,47,51,43,50,53,49,52);(6,8,11,18,10,19,12,17)(7,13,15,20,9,16,14,21)(22,24)(23,29,32,44)(25,34)(26,35)(27,37)(28,36)(30,43,38,52)(31,33,42,48)(39,49)(40,51)(41,50)(45,53,46,47);(6,9,10,7)(8,16,19,13)(11,14,12,15)(17,20,18,21)(22,25,28,41)(23,30,32,38)(24,34,36,50)(26,40,27,39)(29,43,44,52)(31,46,42,45)(33,47,48,53)(35,51,37,49)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2360);