Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1705 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #127 |
| commutator, Frattini, cosocle | C42 | C23 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,5)(3,6)(7,8)(10,13)(11,14)(15,16);(1,2,6,8,4,7,3,5);(9,10,14,16,12,15,11,13)Polycyclic presentation:
2^7|b^2=df,c^2=eg,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg38 conjugacy classes — view the full character table.
| Frattini subgroup of | #94819, #94823, #94824, … |
| Fitting subgroup of | #1712⋊C3, #11710973, #11711452 |
| maximal subgroup of | #7096, #7529, #12310, … |
| maximal normal subgroup of | #7096, #7529, #12310, … |
| maximal quotient of | #8818, #9824, #10304, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 444);