Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#13384 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C5×#3061 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(7,8)(10,11)(12,16)(13,17)(14,19)(15,21)(18,26)(20,27)(22,35)(23,32)(24,30)(25,31)(28,34)(29,36)(33,39)(37,41)(38,40);(6,7)(8,9)(10,12)(11,16)(13,23)(14,22)(15,25)(17,32)(18,29)(19,35)(20,30)(21,31)(24,27)(26,36)(28,37)(33,40)(34,41)(38,39);(6,8,9,7)(11,17)(12,22)(13,26)(14,27)(16,29)(18,20)(19,33)(21,34)(23,37)(24,25)(28,39)(30,41)(31,40)(32,35)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6372);