Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C11 · Schur multiplier (H2) = C29 |
| direct product | C11×#3573 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C11 | C24 |
| commutator | C2 | C24×C4×C11 |
| Frattini, cosocle | C4 | C25×C11 |
| socle | C2×C11 | C24×C4 |
| prime radical | C11×#585 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C11 ◁ G | |
1424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 5 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,16,20,17,21,27,33)(14,18,24,30,23,29,36,40)(15,19,26,32,25,31,37,41)(22,28,35,39,34,38,42,43);(12,14)(13,18)(15,22)(16,24)(17,23)(19,28)(20,30)(21,29)(25,34)(26,35)(27,36)(31,38)(32,39)(33,40)(37,42)(41,43);(12,15)(13,19)(14,22)(16,26)(17,25)(18,28)(20,32)(21,31)(23,34)(24,35)(27,37)(29,38)(30,39)(33,41)(36,42)(40,43);(15,25)(19,31)(22,34)(26,37)(28,38)(32,41)(35,42)(39,43);(14,23)(18,29)(22,34)(24,36)(28,38)(30,40)(35,42)(39,43)748 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 2305);