Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#2022 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | #120 |
| commutator, Frattini, cosocle | C24 | C23×C11 |
| prime radical | C11×#534 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C24×C11 ◁ G | |
430 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 430 conjugacy classes of subgroups — too many to draw. Only its 92 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)(12,13)(14,15)(16,17)(18,21)(19,22)(20,23)(24,27)(25,28)(26,30)(29,31);(16,18,20,25)(17,21,23,28)(19,24,26,31)(22,29,30,27);(13,15)(16,19)(17,22)(18,24)(20,26)(21,27)(23,30)(25,31)(28,29)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 754);