Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C4×AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator | C4 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| socle | C22 | C22×D4 |
| prime radical | C2×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
274 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 12 points:
(1,2,3,4);(5,6)(7,9)(8,10)(11,12);(6,9)(7,12)(10,11);(6,10)(9,11)44 conjugacy classes — view the full character table.
| maximal subgroup of | #8954, #8956, #9095, … |
| maximal normal subgroup of | #8954, #8956, #9095, … |
| maximal quotient of | C8×AGL(1,ℤ/8), #17869, #17871, … |
| automorphism group of | C5×D8, C5×Q16 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1676);