Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#11031 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #392 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11)(12,21)(13,26,29,18)(14,30)(15,19)(16,20)(17,23)(22,34,39,38)(24,33)(25,37)(27,41,40,32)(28,35,31,36);(7,9)(10,12,16,25)(11,17,20,33)(13,27,29,40)(14,24,15,23)(18,34,26,38)(19,37,30,21)(22,31,39,28)(32,35,41,36);(10,13)(11,18)(12,22)(14,28)(15,31)(16,29)(17,32)(19,35)(20,26)(21,38)(23,27)(24,40)(25,39)(30,36)(33,41)(34,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4019);