Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator | C4 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| socle | C22 | C22×D4 |
| prime radical | C2×#135 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,14)(8,13)(9,17)(10,12)(11,15)(16,20)(18,19);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,20)(14,17);(1,3,4,2)(6,12)(11,19)(13,17)(14,20);(5,8,10,9)(6,13,12,17)(7,15,16,18)(11,20,19,14)Polycyclic presentation:
2^7|a^2=f,c^2=f,d^2=g,e^2=g,b^a=be,c^a=cg,e^a=eg,e^b=eg44 conjugacy classes — view the full character table.
| maximal subgroup of | #12649, #12943, #13025, … |
| maximal normal subgroup of | #12649, #12943, #13025, … |
| maximal quotient of | #17849, #17868, #17890, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1675);