Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(7,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C11 · Schur multiplier (H2) = C214 |
| direct product | C11×2−1+6 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | C26 |
| commutator, Frattini, cosocle | C2 | C26×C11 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ G | |
5796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 6 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,19)(15,20)(16,28)(17,18)(21,29)(22,32)(23,37)(24,30)(25,34)(26,40)(27,31)(33,38)(35,41)(36,43)(39,42);(13,18)(19,30)(20,31)(21,28)(32,34)(33,37)(35,40)(42,43);(12,14,17,24)(13,19,18,30)(15,25,27,22)(16,23,29,38)(20,34,31,32)(21,33,28,37)(26,39,41,36)(35,43,40,42);(12,15,17,27)(13,20,18,31)(14,22,24,25)(16,26,29,41)(19,32,30,34)(21,35,28,40)(23,36,38,39)(33,42,37,43);(12,13)(14,19)(15,20)(16,21)(17,18)(22,32)(23,33)(24,30)(25,34)(26,35)(27,31)(28,29)(36,42)(37,38)(39,43)(40,41);(12,16)(13,21)(14,23)(15,26)(17,29)(18,28)(19,33)(20,35)(22,36)(24,38)(25,39)(27,41)(30,37)(31,40)(32,42)(34,43)715 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 2327);