Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = #72232191 (order 2048) · Inn(G) = C2×D4 · Out(G) = C2×D42 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C22 | C2×C22⋊C4 |
| prime radical | C22×#41 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
309 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 309 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 32 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2)(3,4)(5,6,8,11,9,7,12,10);(5,7)(6,9)(8,10)(11,12);(2,4)Polycyclic presentation:
2^7|a^2=f,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,d^b=dg38 conjugacy classes — view the full character table.
| maximal subgroup of | #8359, #8361, #8364, … |
| maximal normal subgroup of | #8359, #8361, #8364, … |
| maximal quotient of | #8310, #8311, #8312, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 521);