Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#10747 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C23 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #379 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
1214 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)(10,11,14,19,15,18,27,32)(12,20,23,33,22,16,26,29)(13,17,25,30,24,31,37,41)(21,28,36,39,35,34,38,40);(6,7,8,9)(10,12)(11,16)(13,21)(14,26)(15,22)(17,28)(18,20)(19,33)(23,27)(24,35)(25,38)(29,32)(30,40)(31,34)(36,37)(39,41);(10,13)(11,17)(12,21)(14,25)(15,24)(16,28)(18,31)(19,30)(20,34)(22,35)(23,36)(26,38)(27,37)(29,39)(32,41)(33,40)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3735);