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) = C24×C4 |
| unambiguous semidirect product | C4⋊#95 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C23 |
| commutator | C22 | C2×C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,4)(6,7)(13,14,16,15);(5,6,8,7)(9,10,11,12)(13,15,16,14);(1,2,3,4)(5,7,8,6)Polycyclic presentation:
2^7|a^2=f,b^2=g,c^2=e,b^a=bd,c^a=ce56 conjugacy classes — view the full character table.
| derived subgroup of | #72267798, #72267844 |
| Frattini subgroup of | #93895, #93896, #93905, … |
| maximal subgroup of | #7025, #7039, #7189, … |
| maximal normal subgroup of | #7025, #7039, #7189, … |
| maximal quotient of | #7554, #7557, #10832, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 167);