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) = C26 |
| direct product | C5×#20888 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C24 | C24×C5 |
| socle | C22×C5 | C2×#120 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
1502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5)(6,7,9,8)(15,21)(22,25);(6,8,9,7)(10,11)(12,15)(13,16)(14,17)(18,22)(19,21)(20,23)(24,25);(10,12)(11,15)(13,18)(14,19)(16,22)(17,21)(20,24)(23,25);(10,13,14,20)(11,16,17,23)(12,18,19,24)(15,22,21,25)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13876);