Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C11 · Schur multiplier (H2) = C24×C4 |
| direct product | C11×#2273 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C11 | C24 |
| commutator | C2 | C22×C42×C11 |
| Frattini, cosocle | C23 | C24×C11 |
| prime radical | C25×C11 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C11 ◁ G | |
912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(16,17,19,18)(20,21)(22,26)(23,24)(25,27);(12,13,14,15)(16,18,19,17)(21,24)(25,26);(20,21)(22,25)(23,24)(26,27);(20,22)(21,25)(23,27)(24,26)748 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1005);