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) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D4 |
| commutator | C4 | C2×C42 |
| Frattini, cosocle | C22×C4 | C23 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,4,3)(9,10,12,14,13,15,11,16);(5,6,7,8)(10,14)(11,12)(15,16);(1,3,4,2)(10,15)(14,16)Polycyclic presentation:
2^7|a^2=e,b^2=f,c^2=de,d^2=g,e^2=g,b^a=bd,c^a=cg,d^b=dg,e^b=eg44 conjugacy classes — view the full character table.
| maximal subgroup of | #12650, #12677, #13119, … |
| maximal normal subgroup of | #12650, #12677, #13119, … |
| maximal quotient of | #8275, #8276, #8279, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 508);