Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33600 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C22×C5 | C22×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G | |
710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
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)(15,30)(16,20)(17,24)(19,22)(23,34)(25,40)(26,36)(27,33)(29,41)(31,38)(32,39)(35,37);(10,12,22,37,15,25,29,39,16,26,27,38,13,23,14,24)(11,17,28,34,18,31,33,36,20,32,41,40,30,35,19,21);(7,9);(10,13,16,15)(11,18,20,30)(12,23,26,25)(14,27,29,22)(17,31,32,35)(19,28,33,41)(21,34,36,40)(24,38,39,37)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26588);