Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) = #72237656 (order 4096) · Inn(G) = C2×D4 · Out(G) = #8534 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator | C4 | C2×C42 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C22 | C2×C4⋊C4 |
| prime radical | C23 | C2×Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,4,3)(5,6,9,14,10,12,8,13)(7,15,17,11,18,20,16,19);(5,7,8,17,10,18,9,16)(6,11,13,20,12,19,14,15);(1,3,4,2)(6,12)(11,19)(13,14)(15,20)Polycyclic presentation:
2^7|a^2=e,b^2=f,c^2=de,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=cg,d^b=dg,e^b=eg44 conjugacy classes — view the full character table.
| maximal subgroup of | #12651, #12678, #13114, … |
| maximal normal subgroup of | #12651, #12678, #13114, … |
| maximal quotient of | #8275, #8276, #8277, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 510);