Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C210 |
| direct product | C5×#60368 |
| 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 ◁ C22×C5×#41 ◁ G | |
4726 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)(12,20)(13,17)(14,26)(15,18)(16,22)(19,28)(21,34)(23,33)(24,31)(25,38)(27,30)(29,36)(32,37)(35,41)(39,40);(10,12,15,23)(11,16,18,30)(13,21,24,35)(14,27,28,22)(17,29,31,40)(19,33,26,20)(25,39,37,36)(32,41,38,34);(7,9)(10,13)(11,17)(12,21)(14,25)(15,24)(16,29)(18,31)(19,32)(20,34)(22,36)(23,35)(26,38)(27,39)(28,37)(30,40)(33,41);(13,24)(17,31)(21,35)(25,37)(29,40)(32,38)(34,41)(36,39);(11,18)(16,30)(17,31)(19,26)(20,33)(29,40)(32,38)(34,41)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53356);