Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | Aut(G) = #72236189 (order 1024) · Inn(G) = #386 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #386 |
| 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 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
173 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 173 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:
(1,2,5,8,6,10,4,9)(3,11,13,15,14,16,12,7);(1,3)(2,7,10,15)(4,12)(5,13)(6,14)(8,11,9,16);(2,8)(4,5)(7,11)(9,10)(12,13)(15,16)Polycyclic presentation:
2^7|a^2=f,b^2=df,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^b=eg,d^c=dg,e^c=eg,f^c=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1904, #20998, #22284, … |
| maximal normal subgroup of | C2×#1904, #20998, #22284, … |
| maximal quotient of | #9087, #9089, #9091, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 636);