Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C11 · Schur multiplier (H2) = C25 |
| direct product | C2×C11×#440 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C11 | C2×D4 |
| commutator | C4 | C23×C4×C11 |
| Frattini, cosocle | C2×C4 | C24×C11 |
| prime radical | C22×D4×C11 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C11 ◁ C2×C42×C11 ◁ G | |
544 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2)(15,16,17,18,19,20,21,22,23,24,25);(3,4,6,5)(7,8)(9,11)(10,12)(13,14);(8,11)(9,14)(12,13);(3,5,6,4)(8,12)(11,13)484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1672);