Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#10764 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×C22⋊C4 |
| commutator | C23 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | C4×C22⋊C4 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C5×C8 ◁ G | |
1194 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)(12,20,22,16)(13,24,25,17)(14,19)(15,18)(21,28)(23,33,36,29)(26,37,38,30)(27,31)(32,35)(34,39)(40,41);(7,9)(10,12,14,23,15,22,27,36)(11,16,19,29,18,20,31,33)(13,21,26,34,25,35,38,41)(17,28,30,39,24,32,37,40);(10,13)(11,17)(12,21)(14,26)(15,25)(16,28)(18,24)(19,30)(20,32)(22,35)(23,34)(27,38)(29,39)(31,37)(33,40)(36,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3752);