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×H(7,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | C26 |
| commutator, Frattini, cosocle | C2 | C26×C11 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ G | |
5992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 6 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,18)(15,22)(16,17)(19,23)(20,24)(21,27)(25,26);(13,17)(18,24)(19,22)(25,27);(14,20)(18,24)(21,26)(25,27);(12,14)(13,18)(15,21)(16,20)(17,24)(19,25)(22,27)(23,26);(12,13)(14,18)(15,19)(16,17)(20,24)(21,25)(22,23)(26,27);(12,15)(13,19)(14,21)(16,23)(17,22)(18,25)(20,26)(24,27)715 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 2326);