Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) = C2×#33468 (order 512) · Inn(G) = D8 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | D8 |
| commutator | C8 | C2×C8 |
| Frattini, cosocle | C4×C8 | C22 |
| socle | C2 | #335 |
| prime radical | C22 | #108 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
31 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,10,6,12,4,20,7,13,15,9,21,11,22,27)(3,14,17,24,18,29,16,8,19,23,30,25,31,28,32,26);(1,3,15,30,6,18,22,32,7,19,5,17,21,31,4,16)(2,8,20,28,11,24,10,23,13,26,27,29,12,25,9,14)Polycyclic presentation:
2^7|a^2=d,b^2=ce,c^2=eg,d^2=f,e^2=g,f^2=g,b^a=bc,c^a=ce,e^a=eg44 conjugacy classes — view the full character table.
| maximal subgroup of | #7387, #7388, #7447, … |
| maximal normal subgroup of | #7387, #7388, #7447, … |
| maximal quotient of | #7100, #7101, #7160, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 105);