Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C23 |
| direct product | C11×#2030 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | #120 |
| commutator, Frattini, cosocle | C24 | C23×C11 |
| prime radical | C24×C11 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C24×C11 ◁ G | |
290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,15,17)(14,18,19,16)(20,21)(22,25)(23,26)(24,27)(28,33)(29,32)(30,34)(31,35);(12,14,15,19)(13,16,17,18)(20,22)(21,25)(23,28)(24,29)(26,33)(27,32)(30,35)(31,34);(20,23,24,30)(21,26,27,34)(22,28,29,35)(25,31,32,33)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 762);