Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#2230 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #476 |
| commutator, Frattini, cosocle | C2×C8 | C23×C11 |
| prime radical | C11×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×C4×C11 ◁ C4×C8×C11 ◁ G | |
126 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 176 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(13,16)(14,22,24,19)(15,17)(20,21)(23,25,26,27);(12,13,15,16,18,21,17,20)(14,19,23,25,24,22,26,27);(12,14)(13,19)(15,23)(16,25)(17,26)(18,24)(20,27)(21,22)253 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 962);