Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | Aut(G) = #72236721 (order 2048) · Inn(G) = #386 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
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 | |
172 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)(3,9)(4,12,14,7)(5,11)(6,8)(10,13,16,15);(1,3,9,2,6,11,5,8)(4,10,13,7,14,16,15,12);(1,4)(2,7)(3,10)(5,15)(6,14)(8,12)(9,13)(11,16)Polycyclic presentation:
2^7|a^2=f,b^2=dfg,d^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 | #12755, #12758, #12778, … |
| maximal normal subgroup of | #12755, #12758, #12778, … |
| maximal quotient of | #9101, #9102, #9105, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 642);