Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33585 |
| 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 ◁ C5×#117 ◁ C5×#405 ◁ G | |
748 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)(7,9)(10,11)(12,16)(13,17)(14,27)(15,20)(18,24)(19,25)(21,29)(22,28)(23,37)(26,33)(30,36)(31,35)(32,38)(34,39)(40,41);(6,7,8,9)(10,12)(11,16)(13,22)(14,23)(15,21)(17,28)(18,30)(19,31)(20,29)(24,36)(25,35)(26,34)(27,37)(32,40)(33,39)(38,41);(11,17)(13,24)(14,25)(16,28)(18,19)(20,33)(22,36)(23,35)(26,38)(27,32)(29,39)(30,31)(34,41)(37,40);(12,21)(16,29)(22,34)(23,35)(28,39)(30,40)(31,37)(36,41)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26573);