Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C2×#72236376 (order 4096) · Inn(G) = C23 · Out(G) = C24×H(5,𝔽2) |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C23 |
| commutator | C22 | C22×C8 |
| socle | C23 | C22×C4 |
| prime radical | C24 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
200 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,4)(5,6)(7,8)(9,10,11,13,12,14,15,16);(1,2,3,4);(6,8)Polycyclic presentation:
2^7|a^2=f,b^2=d,f^2=g,b^a=bd,c^a=ce56 conjugacy classes — view the full character table.
| maximal subgroup of | #7020, #7024, #7028, … |
| maximal normal subgroup of | #7020, #7024, #7028, … |
| maximal quotient of | #8557, #8558, #8559, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 529);