Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | Aut(G) = #72236671 (order 6144) · Inn(G) = #120 · Out(G) = D4×S4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C24 | C23 |
| prime radical | C25 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
233 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,4,7)(3,9,10,6)(5,8,12,14)(11,15,16,13);(1,3,4,10)(2,6,7,9)(5,11,12,16)(8,13,14,15);(6,13)(9,15)Polycyclic presentation:
2^7|a^2=d,b^2=d,b^a=bd,c^a=ce,f^a=fg,c^b=cf,e^b=eg26 conjugacy classes — view the full character table.
| Fitting subgroup of | #2029⋊C3, #11710965, #11711401, … |
| maximal subgroup of | #7426, #12202, #12203, … |
| maximal normal subgroup of | #7426, #12202, #12203, … |
| maximal quotient of | #10340, #10341, #10344, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 761);