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 | C2×C11×#568 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | C24 |
| commutator | C2 | C24×C4×C11 |
| Frattini, cosocle | C4 | C25×C11 |
| socle | C22×C11 | C23×C4 |
| prime radical | C22×C11×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ G | |
1460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29);(3,4,6,10,7,9,13,16)(5,8,12,15,11,14,17,18);(3,5)(4,8)(6,12)(7,11)(9,14)(10,15)(13,17)(16,18);(5,11)(8,14)(12,17)(15,18);(4,9)(8,14)(10,16)(15,18)748 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 2304);