Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | Aut(G) = #72213504 (order 1024) · Inn(G) = C2×#99 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#99 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C2×D4 | C23 |
| prime radical | #535 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
143 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,9,10,16)(4,12,14,7)(5,8)(6,11)(13,15);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14);(3,10)(6,15)(9,16)(11,13)Polycyclic presentation:
2^7|a^2=e,d^2=g,b^a=bd,d^a=df,f^a=fg,c^b=cg,d^b=dg,e^b=efg,e^d=eg23 conjugacy classes — view the full character table.
| maximal subgroup of | #12707, #12708, #12727, … |
| maximal normal subgroup of | #12707, #12708, #12727, … |
| maximal quotient of | #11432, #11448, #11479, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 859);