Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9279 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9280 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C23×C5 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
382 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)(7,11,13,16)(8,14,15,17)(22,23)(24,32)(25,36,38,29)(26,39)(27,31)(28,34)(30,41)(33,46,49,53)(35,47)(37,44,40,45)(42,48,52,50)(43,51);(6,7,9,16,10,13,18,11)(8,12,17,20,15,21,14,19)(22,24,26,34,27,35,41,51)(23,28,30,32,31,43,39,47)(25,33,40,50,38,49,37,48)(29,42,45,46,36,52,44,53);(6,8)(7,12)(9,17)(10,15)(11,19)(13,21)(14,18)(16,20)(22,25,27,38)(23,29,31,36)(24,33,35,49)(26,40,41,37)(28,42,43,52)(30,45,39,44)(32,46,47,53)(34,50,51,48)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2268);