Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | Aut(G) = #72225648 (order 2048) · Inn(G) = #120 · Out(G) = D42 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator | C23 | C22×C4 |
| Frattini, cosocle | C24 | C23 |
| prime radical | C2×#120 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
289 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 289 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) = 12 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 12 points:
(2,4)(5,6)(7,11,12,9)(8,10);(1,2,3,4)(5,6)(7,9)(8,10)(11,12);(5,7)(6,9)(8,12)(10,11)Polycyclic presentation:
2^7|a^2=f,b^2=df,b^a=bd,c^a=ce,e^a=eg,e^b=eg,d^c=dg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #12753, #12759, #12761, … |
| maximal normal subgroup of | #12753, #12759, #12761, … |
| maximal quotient of | #8965, #8967, #9009, … |
| outer automorphism group of | #5067, #5069 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 621);