Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) = C2×#72237160 (order 2048) · Inn(G) = C2×D8 · Out(G) = C22×#41 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
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 | |
71 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10,13,17)(11,14,20,23)(12,21,19,16)(15,24,22,18);(1,3,4,8)(2,5,6,7);(10,14)(11,18)(12,19)(15,16)(17,23)(20,24)(21,22)Polycyclic presentation:
2^7|a^2=dg,b^2=d,e^2=fg,f^2=g,b^a=bd,c^a=ce,e^a=ef,f^a=fg,e^c=ef,f^c=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | #12372, #12375, #12378, … |
| maximal normal subgroup of | #12372, #12375, #12378, … |
| maximal quotient of | #12811, #12937, #13111, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 958);