Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | Aut(G) = C2×#21452 (order 512) · Inn(G) = #102 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #102 |
| commutator | C2×C8 | C2×C4 |
| Frattini, cosocle | C22×C8 | C22 |
| prime radical | #466 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
57 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,17,14,19,13,18)(11,20,22,16,23,15,21,24);(1,3)(2,5)(4,8)(6,7)(10,15)(11,21)(12,13)(16,17)(18,24)(19,20)(22,23)Polycyclic presentation:
2^7|a^2=d,c^2=fg,d^2=g,e^2=g,f^2=g,b^a=bc,c^a=ce,e^a=eg,f^a=fg,c^b=cf,d^b=defg,e^b=eg,f^b=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1349, #11709, #11710, … |
| maximal normal subgroup of | C2×#1349, #11709, #11710, … |
| maximal quotient of | #7080, #7083, #7084, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 81);