Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) = C26×D4 (order 512) · Inn(G) = C2×D4 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | SD16⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator | C4 | C22×C8 |
| Frattini, cosocle | C42 | C23 |
| socle | C22 | C4×D4 |
| prime radical | C2×D4 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,5)(4,6)(7,8)(9,10,12,15,13,11,14,16);(2,5)(3,8)(6,7);(2,6)(5,7)(9,11,12,16,13,10,14,15)Polycyclic presentation:
2^7|a^2=e,c^2=e,d^2=f,e^2=g,b^a=bd,c^a=cf,d^a=df,d^b=df44 conjugacy classes — view the full character table.
| maximal subgroup of | C2×SD16⋊C8, #17849, C8×AGL(1,ℤ/8), … |
| maximal normal subgroup of | C2×SD16⋊C8, #17849, C8×AGL(1,ℤ/8), … |
| maximal quotient of | #8200, #8201, #8285, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 310);