Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
172 conjugacy classes of subgroups.
| 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)(10,13)(11,15)(14,16);(6,8)(9,10,11,13,12,14,15,16);(1,2,3,4)(10,14)(13,16)Polycyclic presentation:
2^7|a^2=f,b^2=df,c^2=e,d^2=g,b^a=bd,c^a=ce,d^a=dg,c^b=cg32 conjugacy classes — view the full character table.
| maximal subgroup of | #7508, #8506, #8507, … |
| maximal normal subgroup of | #7508, #8506, #8507, … |
| maximal quotient of | #9640, #9646, #9682, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 691);