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×#33587 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#3155 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ C5×#405 ◁ G | |
712 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)(12,20)(13,23)(14,25)(16,28)(18,33)(19,32)(21,38)(22,37)(24,27)(26,36)(29,40)(30,41)(31,39)(34,35);(6,7,8,9)(10,11)(12,16)(13,18)(14,19)(15,17)(20,28)(21,30)(22,29)(23,33)(24,34)(25,32)(26,31)(27,35)(36,39)(37,40)(38,41);(10,12,15,22)(11,16,17,29)(13,24,26,38)(14,27,25,21)(18,34,31,41)(19,35,32,30)(20,36,37,23)(28,39,40,33);(11,17)(16,29)(18,31)(19,32)(28,40)(30,35)(33,39)(34,41)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26575);