Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2204 |
| automorphism | Aut(G) = #499787 (order 512) · Inn(G) = B(4,𝔽2) · Out(G) = D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C23 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
118 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)(4,7)(5,13)(6,10)(8,12)(11,16)(14,15);(2,7)(3,5)(4,12)(8,10)(9,15)(13,16);(1,3)(4,11)(5,6)(7,8)(9,13)(12,14)Polycyclic presentation:
2^7|d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,c^b=cg,d^b=dg,e^b=efg,d^c=df,e^c=eg,e^d=eg20 conjugacy classes — view the full character table.
| maximal subgroup of | #13681, #13682, C2×#2204, … |
| maximal normal subgroup of | #13681, #13682, C2×#2204, … |
| maximal quotient of | #12099, #12100, #12129, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 936);