Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | Aut(G) = #499787 (order 512) · Inn(G) = B(4,𝔽2) · Out(G) = D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C23 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 8 |
A minimal faithful permutation representation — 3 generators on 8 points:
(1,2)(3,5)(4,6)(7,8);(2,5)(6,8);(3,7)Polycyclic presentation:
2^7|b^a=bd,c^a=ce,f^a=fg,e^b=efg,d^c=df,e^d=eg20 conjugacy classes — view the full character table.
| maximal subgroup of | #13673, #13674, C2×#2196, … |
| maximal normal subgroup of | #13673, #13674, C2×#2196, … |
| maximal quotient of | #12090, #12096, #12120, … |
| automorphism group of | #100, #101 |
| outer automorphism group of | #534, #537, #544, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 928);