Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) = #72235124 (order 2048) · Inn(G) = C2×D4 · Out(G) = #3484 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
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 | C4×D4 |
| prime radical | C2×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,9,13,10,14,8,11)(7,15,18,20,16,19,17,12);(1,2,3,4)(6,11,14,13)(7,16)(12,15,20,19)(17,18);(5,7,8,17,10,16,9,18)(6,12,13,15,14,20,11,19)Polycyclic presentation:
2^7|a^2=f,b^2=df,c^2=e,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=cg32 conjugacy classes — view the full character table.
| maximal subgroup of | #7109, #13024, #13037, … |
| maximal normal subgroup of | #7109, #13024, #13037, … |
| maximal quotient of | #9658, #9659, #9660, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 707);