Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | Aut(G) = #7622868 (order 512) · Inn(G) = #99 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #99 |
| commutator | C23 | C2×C8 |
| Frattini, cosocle | C2×M16 | C22 |
| socle | C2 | #323 |
| prime radical | C24 | C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,7,4,8,10,14,5,6,9,12,11,13,15,16);(2,6)(8,13)Polycyclic presentation:
2^7|a^2=d,d^2=f,f^2=g,b^a=bc,c^a=ce,e^a=eg,d^b=de,d^c=dg26 conjugacy classes — view the full character table.
| maximal subgroup of | #7340, #7341, #7342, … |
| maximal normal subgroup of | #7340, #7341, #7342, … |
| maximal quotient of | #7054, #7055, #7060, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 52);