Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28×C4 |
| direct product | C5×#60649 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×#41 ◁ G | |
3178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 41 points:
(1,2,3,4,5)(7,9)(10,11)(12,20)(13,18)(14,26)(15,17)(16,22)(19,27)(21,33)(23,32)(24,30)(25,38)(28,36)(29,35)(31,37)(34,40)(39,41);(10,12,14,22,15,23,27,36)(11,16,19,20,17,28,26,32)(13,21,25,35,24,34,37,41)(18,29,31,33,30,39,38,40);(6,7,8,9)(11,17)(16,28)(18,30)(19,26)(20,32)(29,39)(31,38)(33,40);(10,13)(11,18)(12,21)(14,25)(15,24)(16,29)(17,30)(19,31)(20,33)(22,35)(23,34)(26,38)(27,37)(28,39)(32,40)(36,41);(13,24)(18,30)(21,34)(25,37)(29,39)(31,38)(33,40)(35,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53637);