Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) = C210 (order 1024) · Inn(G) = C23 · Out(G) = C27 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C23 |
| commutator, socle | C23 | C22×C4 |
| prime radical | C24 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
140 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 60 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:
(1,2)(3,4)(6,7)(9,10,13,15,12,11,14,16);(5,6,8,7)(9,11,13,16,12,10,14,15);(2,4)(5,7,8,6)(10,11)(15,16)Polycyclic presentation:
2^7|a^2=f,b^2=df,c^2=d,f^2=g,b^a=bd,c^a=ce,c^b=cg44 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1924, #21047, #21048, … |
| maximal normal subgroup of | C2×#1924, #21047, #21048, … |
| maximal quotient of | #9545, #9546, #9547, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 656);