Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C27 |
| direct product | C2×C5×#1890 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | #120 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C24 | C24×C5 |
| prime radical | C23×C5×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C25×C5 ◁ G | |
2102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(23,24,25,26,27);(4,6)(7,8)(9,12)(10,17,18,13)(11,14)(15,19,21,22)(16,20);(3,4,5,6)(7,9,11,16)(8,12,14,20)(10,15,18,21)(13,19,17,22);(7,10)(8,13)(9,15)(11,18)(12,19)(14,17)(16,21)(20,22)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13946);