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×#10748 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C23 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #385 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
1190 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)(10,11,14,20,16,19,15,21)(12,22,25,35,24,17,29,31)(13,18,27,32,26,34,28,33)(23,30,38,40,37,36,39,41);(10,12,15,25,16,24,14,29)(11,17,21,31,19,22,20,35)(13,23,28,38,26,37,27,39)(18,30,33,40,34,36,32,41);(7,9)(10,13)(11,18)(12,23)(14,27)(15,28)(16,26)(17,30)(19,34)(20,32)(21,33)(22,36)(24,37)(25,38)(29,39)(31,40)(35,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3736);