Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C210 |
| direct product | C5×#60366 |
| 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 ◁ C22×C5×#41 ◁ G | |
4782 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)(6,7)(8,9)(10,11)(12,21)(13,18)(14,19)(15,28)(16,20)(17,23)(22,35)(24,31)(25,33)(26,34)(27,38)(29,32)(30,37)(36,41)(39,40);(10,12)(11,17)(13,22)(14,23)(15,29)(16,24)(18,30)(19,31)(20,32)(21,28)(25,36)(26,37)(27,39)(33,40)(34,41)(35,38);(7,9)(10,13)(11,18)(12,22)(14,26)(15,27)(16,25)(17,30)(19,34)(20,33)(21,35)(23,37)(24,36)(28,38)(29,39)(31,41)(32,40);(13,25)(18,33)(22,36)(26,27)(30,40)(34,38)(35,41)(37,39);(10,14,16,15)(11,19,20,28)(12,23,24,29)(13,26,25,27)(17,31,32,21)(18,34,33,38)(22,37,36,39)(30,41,40,35)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53354);