Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C11 · Schur multiplier (H2) = C29 |
| direct product | C4×C11×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | C24 |
| commutator | C2 | C24×C4×C11 |
| Frattini, cosocle | C22 | C25×C11 |
| socle | C22×C11 | C25 |
| prime radical | C2×C11×H(5,𝔽2) | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ G | |
1660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2,3,4)(13,14,15,16,17,18,19,20,21,22,23);(5,6)(7,11)(8,9)(10,12);(6,9)(10,11);(5,6)(7,10)(8,9)(11,12);(5,7)(6,10)(8,12)(9,11)748 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 2161);