Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28×C4 |
| direct product | C5×#60651 |
| 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 | |
3138 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)(10,11,15,19)(12,20,21,33)(13,17,24,31)(14,26,27,18)(16,28,29,23)(22,34,35,41)(25,38,37,32)(30,39,40,36);(10,12,14,23,15,21,27,28)(11,16,18,20,19,29,26,33)(13,22,25,36,24,35,37,39)(17,30,32,34,31,40,38,41);(6,7,8,9)(12,21)(16,29)(20,33)(22,35)(23,28)(30,40)(34,41)(36,39);(10,13)(11,17)(12,22)(14,25)(15,24)(16,30)(18,32)(19,31)(20,34)(21,35)(23,36)(26,38)(27,37)(28,39)(29,40)(33,41);(13,24)(17,31)(22,35)(25,37)(30,40)(32,38)(34,41)(36,39)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53639);