Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | Aut(G) = C2×#3109 (order 256) · Inn(G) = #442 · Out(G) = C22 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
| affine general linear | AGL(1,ℤ/16) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #442 |
| commutator | C8 | C22×C4 |
| Frattini, cosocle | C2×C8 | C23 |
| prime radical | #509 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
77 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,6,8,15)(3,10)(4,11)(7,13,16,9)(12,14);(1,2)(3,7)(4,13)(5,8)(6,14)(9,11)(10,15)(12,16);(1,3,11,14,5,12,4,10)(2,7,9,6,8,16,13,15)Polycyclic presentation:
2^7|a^2=e,c^2=efg,d^2=f,f^2=g,b^a=bd,c^a=cf,d^a=df,f^a=fg,d^b=dfg,e^b=eg,f^b=fg,d^c=dg23 conjugacy classes — view the full character table.
| maximal subgroup of | C2×AGL(1,ℤ/16), #33468, #33469, … |
| maximal normal subgroup of | C2×AGL(1,ℤ/16), #33468, #33469, … |
| maximal quotient of | #12065, #12067, #12077, … |
| automorphism group of | D16, Q32 |
| holomorph of | C16 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 913);