Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | C22×D8 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,4)(10,15)(11,16)(12,13);(5,6)(7,8)(9,10)(11,12)(13,15)(14,16);(2,4)(6,8)(9,11,12,10,14,15,13,16)Polycyclic presentation:
2^7|c^2=d,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #7080, #7088, #12158, … |
| maximal normal subgroup of | #7080, #7088, #12158, … |
| maximal quotient of | #10472, #10473, #10484, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 786);