Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C11 · Schur multiplier (H2) = C27 |
| direct product | C22×C11×#102 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C11 | D4 |
| commutator | C4 | C23×C4×C11 |
| Frattini, cosocle | C2×C4 | C24×C11 |
| prime radical | C23×D4×C11 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C11 ◁ C23×C4×C11 ◁ G | |
1072 conjugacy classes of subgroups; only the 440 normal subgroups are shown.
This group has 1072 conjugacy classes of subgroups — too many to draw. Only its 440 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(17,18,19,20,21,22,23,24,25,26,27);(3,4);(5,6,7,8)(9,10)(11,13)(12,15)(14,16);(10,13)(11,16)(14,15)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1622);