Finite Group DatabasebetaFinite Group Databasebeta| order | 448 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C25 |
| direct product | C2×C7×AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C24×C7 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ G | |
298 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 298 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,2)(3,4,5,6,7,8,9);(10,11)(12,14)(13,15)(16,17);(11,14)(12,17)(15,16);(1,2)(11,15)(14,16)154 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(448, 1356);