Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33611 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#41 ◁ C2×C5×#131 ◁ G | |
674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11)(12,21)(13,18)(14,28)(15,27)(16,20)(17,24)(19,29)(22,34)(23,35)(25,41)(26,37)(30,31)(32,38)(33,40)(36,39);(10,12,22,38,15,25,14,24,16,26,29,39,13,23,30,40)(11,17,31,41,18,32,19,21,20,33,28,35,27,36,34,37);(7,9)(10,11)(12,17)(13,27)(14,19)(15,18)(16,20)(21,24)(22,31)(23,36)(25,32)(26,33)(28,29)(30,34)(35,39)(37,40)(38,41);(10,13,16,15)(11,18,20,27)(12,23,26,25)(14,22,30,29)(17,32,33,36)(19,28,34,31)(21,35,37,41)(24,38,40,39)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26599);