Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8491 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#8502 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #379 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C5×#3547 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5)(6,7)(8,15,17,11)(9,14)(10,13)(12,16,20,18)(19,21)(22,23,27,32,28,33,26,31)(24,34,38,29,37,44,36,43)(25,30,39,45,40,46,41,47)(35,49,50,53,51,48,52,42);(6,8)(7,11)(9,16)(10,17)(12,14)(13,15)(18,19)(20,21)(22,24)(23,29)(25,35)(26,38)(27,36)(28,37)(30,42)(31,44)(32,34)(33,43)(39,52)(40,51)(41,50)(45,48)(46,53)(47,49);(6,8)(7,12)(9,18)(10,17)(11,14)(13,20)(15,21)(16,19)(22,25,28,40)(23,30,33,46)(24,35,37,51)(26,41,27,39)(29,42,43,53)(31,47,32,45)(34,48,44,49)(36,52,38,50)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1490);