Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | Aut(G) = #72237166 (order 2048) · Inn(G) = C2×D8 · Out(G) = #407 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| prime radical | C2×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
69 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,23)(12,17)(13,31)(14,34)(15,35)(16,22)(18,27)(19,24)(20,33)(21,32)(25,37)(26,36)(28,40)(29,39)(30,38);(2,5)(6,8)(9,11,24,29,15,26,33,30,16,27,14,25,32,28,13,12)(10,17,31,40,21,37,34,18,22,38,20,36,35,39,19,23);(1,3,4,7)(2,6,5,8)(9,12,13,28,32,25,14,27,16,30,33,26,15,29,24,11)(10,18,19,37,35,40,20,17,22,23,34,39,21,36,31,38)Polycyclic presentation:
2^7|b^2=eg,c^2=d,d^2=fg,e^2=f,f^2=g,b^a=bd,c^a=ce,d^a=df,e^a=efg,f^a=fg38 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2242, #33697, #33702, … |
| maximal normal subgroup of | C2×#2242, #33697, #33702, … |
| maximal quotient of | #12889, #13147, #13148, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 974);