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) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #121 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C2 | #386 |
| prime radical | #575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
166 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,6,10,16)(4,9)(5,8)(7,11,15,12)(13,14);(1,3)(2,6,8,16)(4,12)(5,10)(7,14,15,9)(11,13);(1,3)(2,7)(4,12)(5,10)(6,14)(8,15)(9,16)(11,13)Polycyclic presentation:
2^7|a^2=f,b^2=df,e^2=g,b^a=bd,c^a=ce,d^a=dg,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=eg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1906, #21000, #21558, … |
| maximal normal subgroup of | C2×#1906, #21000, #21558, … |
| maximal quotient of | #9095, #9096, #9097, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 638);