Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | Aut(G) = C2×#2196 (order 256) · Inn(G) = AΣL(1,𝔽16) · Out(G) = C22 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C2×C4 |
| Frattini, cosocle | C4⋊Q8 | C22 |
| prime radical | 2−1+4 | C4 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
53 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,4,8,6,3,10,12)(5,9,15,7,16,13,14,11);(1,3)(2,6)(4,7)(5,9)(8,15)(10,11)(12,14)(13,16)Polycyclic presentation:
2^7|a^2=d,c^2=g,d^2=g,e^2=g,b^a=bc,c^a=ce,e^a=ef,f^a=fg,c^b=cg,d^b=de,e^b=eg,f^b=fg,d^c=dfg,e^c=eg,e^d=eg17 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1405, #13572, #13573, … |
| maximal normal subgroup of | C2×#1405, #13572, #13573, … |
| maximal quotient of | #7345, #7376, #7409, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 137);