Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) = C23×#442 (order 512) · Inn(G) = D8 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D8 |
| commutator | C8 | C2×C8 |
| Frattini, cosocle | C4×C8 | C22 |
| socle, prime radical | C22 | #102 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
49 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,5,4,6,7,8)(10,14)(11,17)(12,19)(15,24)(16,23)(18,21)(20,22);(9,10,13,16)(11,18,20,24)(12,21,19,15)(14,22,23,17)Polycyclic presentation:
2^7|a^2=d,b^2=g,c^2=eg,d^2=f,e^2=g,b^a=bc,c^a=ce,e^a=eg,c^b=ce,e^b=eg44 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1332, #11591, #11592, … |
| maximal normal subgroup of | C2×#1332, #11591, #11592, … |
| maximal quotient of | #7069, #7073, #7113, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 64);