Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) = C24×D42 (order 1024) · Inn(G) = #121 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #121 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C2×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
166 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,10,12,9,8,14,17)(7,13,15,20,16,11,18,19);(5,7,14,15,9,16,10,18)(6,11,12,20,8,13,17,19);(2,4)(5,8)(6,9)(7,13)(10,12)(11,16)(14,17)(15,19)(18,20)Polycyclic presentation:
2^7|a^2=f,b^2=df,e^2=g,f^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×#1907, #21003, #21004, … |
| maximal normal subgroup of | C2×#1907, #21003, #21004, … |
| maximal quotient of | #9095, #9097, #9163, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 639);