Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1828 |
| automorphism | Aut(G) = #72237014 (order 4096) · Inn(G) = #381 · Out(G) = AΣL(1,𝔽16) |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #381 |
| commutator | C23 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,4,8,5,6,12,13)(3,9,11,16,10,7,14,15);(2,6)(4,12)(7,9)(11,14);(1,3)(2,7,6,9)(4,11)(5,10)(8,15,13,16)(12,14)Polycyclic presentation:
2^7|a^2=f,c^2=d,f^2=g,b^a=bd,c^a=ce,d^a=dg,f^b=fg,e^c=eg29 conjugacy classes — view the full character table.
| maximal subgroup of | #12725, #12726, C2×#1829, … |
| maximal normal subgroup of | #12725, #12726, C2×#1829, … |
| maximal quotient of | #8734, #8736, #8751, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 561);