Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28×C4 |
| direct product | C5×#60643 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| socle | C22×C5 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×#117 ◁ G | |
3194 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)(12,21)(13,18)(14,19)(15,29)(16,20)(17,24)(22,34)(23,31)(25,35)(26,32)(27,33)(28,39)(30,38)(36,41)(37,40);(10,12,15,24,16,25,14,23)(11,17,19,21,20,31,29,35)(13,22,28,38,26,36,27,37)(18,30,33,34,32,40,39,41);(6,7,8,9)(10,13)(11,18)(12,22)(14,27)(15,28)(16,26)(17,30)(19,33)(20,32)(21,34)(23,37)(24,38)(25,36)(29,39)(31,40)(35,41);(13,26)(18,32)(22,36)(27,28)(30,40)(33,39)(34,41)(37,38);(10,14,16,15)(11,19,20,29)(12,23,25,24)(13,27,26,28)(17,21,31,35)(18,33,32,39)(22,37,36,38)(30,34,40,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53631);