Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) = #72235118 (order 2048) · Inn(G) = C2×D4 · Out(G) = C2×#392 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×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×C22⋊C4 |
| prime radical | C2×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,4,3)(5,6)(7,13)(8,16)(9,12)(10,15)(11,17)(14,19)(18,20);(5,7)(6,10,12,18)(8,15)(9,14)(11,13,16,19)(17,20);(1,3,4,2)(7,14)(10,18)(13,19)(15,20)Polycyclic presentation:
2^7|a^2=e,b^2=f,c^2=e,d^2=g,b^a=bd,d^a=dg,f^a=fg,c^b=cg44 conjugacy classes — view the full character table.
| maximal subgroup of | #8412, #8413, #8414, … |
| maximal normal subgroup of | #8412, #8413, #8414, … |
| maximal quotient of | #8170, #8171, #8174, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 491);