Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | Aut(G) = #72236986 (order 2048) · Inn(G) = #386 · Out(G) = #120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #386 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | #535 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
181 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,5,9,6,8,4,14)(3,10,13,15,12,16,11,7);(1,3)(2,7,8,15)(4,11)(5,13)(6,12)(9,10,14,16);(2,8)(4,5)(7,15)(11,13)Polycyclic presentation:
2^7|a^2=f,b^2=df,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,e^b=eg,d^c=dg,f^c=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | #12753, #12754, C2×#1911, … |
| maximal normal subgroup of | #12753, #12754, C2×#1911, … |
| maximal quotient of | #9099, #9101, #9103, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 643);