Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#1890 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | #120 |
| commutator | C23 | C22×C4×C11 |
| Frattini, cosocle | C24 | C23×C11 |
| prime radical | C22×D4×C11 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C24×C11 ◁ G | |
386 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 386 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 352 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7,8,9,10,11)(13,15)(16,17)(18,21)(19,26,27,22)(20,23)(24,28,30,31)(25,29);(12,13,14,15)(16,18,20,25)(17,21,23,29)(19,24,27,30)(22,28,26,31);(16,19)(17,22)(18,24)(20,27)(21,28)(23,26)(25,30)(29,31)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 622);