Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C11 · Schur multiplier (H2) = C1 |
| direct product | C11×SD128 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | D32 |
| commutator, Frattini, cosocle | C32 | C22×C11 |
| prime radical | C11×D32 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C4×C11 ◁ C8×C11 ◁ C11×C16 ◁ C11×C32 ◁ G | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 2 generators on 75 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75);(13,43)(14,74)(15,41)(16,72)(17,39)(18,70)(19,37)(20,68)(21,35)(22,66)(23,33)(24,64)(25,31)(26,62)(27,29)(28,60)(30,58)(32,56)(34,54)(36,52)(38,50)(40,48)(42,46)(45,75)(47,73)(49,71)(51,69)(53,67)(55,65)(57,63)(59,61)385 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 162);