Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(7,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4×C5 · Schur multiplier (H2) = C214 |
| direct product | C5×#62653 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C26 |
| commutator | C2 | C25×C4×C5 |
| Frattini, cosocle | C22 | C26×C5 |
| prime radical | C22×C5×2−1+4 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
10930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 6 generators on 41 points:
(1,2,3,4,5)(6,7,8,9)(10,11)(12,17)(13,18)(14,26)(15,16)(19,27)(20,30)(21,35)(22,28)(23,32)(24,38)(25,29)(31,36)(33,39)(34,41)(37,40);(11,16)(17,28)(18,29)(19,26)(30,32)(31,35)(33,38)(40,41);(10,12,15,22)(11,17,16,28)(13,23,25,20)(14,21,27,36)(18,32,29,30)(19,31,26,35)(24,37,39,34)(33,41,38,40);(10,13,15,25)(11,18,16,29)(12,20,22,23)(14,24,27,39)(17,30,28,32)(19,33,26,38)(21,34,36,37)(31,40,35,41);(10,11)(12,17)(13,18)(14,19)(15,16)(20,30)(21,31)(22,28)(23,32)(24,33)(25,29)(26,27)(34,40)(35,36)(37,41)(38,39);(10,14)(11,19)(12,21)(13,24)(15,27)(16,26)(17,31)(18,33)(20,34)(22,36)(23,37)(25,39)(28,35)(29,38)(30,40)(32,41)650 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55641);