Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C23 |
| direct product | C2×C11×#355 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | #99 |
| commutator | C2×C4 | C22×C4×C11 |
| Frattini, cosocle | C2×Q8 | C23×C11 |
| prime radical | C22×D4×C11 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C23×C11 ◁ C22×Q8×C11 ◁ G | |
260 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29);(3,4,6,10,7,11,15,18)(5,9,13,17,12,8,14,16);(4,8)(5,12)(9,11)(10,17)(13,14)(16,18)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 862);