Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | Aut(G) = #72236721 (order 2048) · Inn(G) = #386 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #386 |
| commutator | C23 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | #535 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,5,9,6,8,15,3)(4,11,13,16,12,7,14,10);(1,3,5,8,6,9,15,2)(4,10,14,11,12,16,13,7);(1,4)(2,7)(3,10)(5,14)(6,12)(8,11)(9,16)(13,15)Polycyclic presentation:
2^7|a^2=f,b^2=df,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg,e^b=eg,f^b=fg,d^c=dg,f^c=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | #12755, #12756, #12769, … |
| maximal normal subgroup of | #12755, #12756, #12769, … |
| maximal quotient of | #9107, #9109, #9187, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 646);