Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C11 · Schur multiplier (H2) = C24×C4 |
| direct product | C11×#2875 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | C24 |
| commutator | C2 | C22×C42×C11 |
| Frattini, cosocle | C2×C4 | C24×C11 |
| socle | C22×C11 | C23×C4 |
| prime radical | C2×C11×#41 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ G | |
556 conjugacy classes of subgroups; only the 504 normal subgroups are shown.
This group has 556 conjugacy classes of subgroups — too many to draw. Only its 504 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3,4,5,6,7,8,9,10,11)(16,17,19,23,20,21,26,28)(18,22,25,29,24,27,30,31);(12,13,14,15)(17,21)(22,27)(23,28)(29,31);(16,18)(17,22)(19,25)(20,24)(21,27)(23,29)(26,30)(28,31);(18,24)(22,27)(25,30)(29,31)748 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1607);