Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33579 |
| 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 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
718 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)(7,9)(10,11)(12,21)(13,18)(14,26)(15,30)(16,20)(17,23)(19,28)(22,37)(24,38)(25,36)(27,32)(29,40)(31,35)(33,41)(34,39);(6,7,8,9);(10,12)(11,17)(13,22)(14,27)(15,31)(16,24)(18,32)(19,35)(20,34)(21,26)(23,28)(25,39)(29,41)(30,33)(36,37)(38,40);(10,13,16,15)(11,18,20,30)(12,22,24,31)(14,25,29,28)(17,32,34,33)(19,26,36,40)(21,37,38,35)(23,27,39,41)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26567);