Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) = C2×#63011 (order 512) · Inn(G) = #121 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #121 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C2 | #388 |
| prime radical | C2×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,5,9,6,10,4,7)(3,11,14,15,13,8,12,16);(2,7,10,9)(8,15,11,16);(1,3,4,12,6,13,5,14)(2,8,9,16,10,11,7,15)Polycyclic presentation:
2^7|a^2=f,b^2=df,c^2=d,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^c=eg32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1932, #21080, #21558, … |
| maximal normal subgroup of | C2×#1932, #21080, #21558, … |
| maximal quotient of | #9549, #9550, #9551, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 664);