Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
| unambiguous semidirect product | C22⋊#95 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C23 |
| commutator | C22 | C2×C42 |
| prime radical | C25 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
544 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)(10,12);(9,10)(11,12)(13,14,15,16);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)Polycyclic presentation:
2^7|a^2=f,b^2=g,b^a=bd,c^a=ce56 conjugacy classes — view the full character table.
| derived subgroup of | #72238057, #72238059, #72267756, … |
| Frattini subgroup of | #93892, #93901, #93909, … |
| maximal subgroup of | #7032, #7035, #7187, … |
| maximal normal subgroup of | #7032, #7035, #7187, … |
| maximal quotient of | #7554, #7555, #7557, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 165);