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) = C23×C4 |
| direct product | C5×#10920 |
| 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 | C24×C5 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
698 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)(10,11,14,18,15,19,25,31)(12,16,22,28,21,27,35,39)(13,17,24,30,23,29,36,40)(20,32,34,41,33,26,37,38);(6,7,8,9)(10,12)(11,16)(13,20)(14,22)(15,21)(17,26)(18,28)(19,27)(23,33)(24,34)(25,35)(29,32)(30,38)(31,39)(36,37)(40,41);(10,13,14,24,15,23,25,36)(11,17,18,30,19,29,31,40)(12,20,22,34,21,33,35,37)(16,26,28,38,27,32,39,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3908);