Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | Aut(G) = #72232993 (order 2048) · Inn(G) = C2×C22⋊C4 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×C22⋊C4 |
| commutator | C23 | C22×C4 |
| Frattini, cosocle | C24 | C23 |
| prime radical | C2×#121 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
204 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 204 conjugacy classes of subgroups — too many to draw. Only its 58 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:
(2,4)(5,6)(7,9,11,12)(8,10);(5,7)(6,9,10,12)(8,11);(1,2,3,4)Polycyclic presentation:
2^7|a^2=f,b^2=df,c^2=e,b^a=bd,c^a=ce,d^a=dg,d^b=dg,f^b=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #8465, #8466, #8479, … |
| maximal normal subgroup of | #8465, #8466, #8479, … |
| maximal quotient of | #9649, #9650, #9651, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 696);