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) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, prime radical, cosocle | C24 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
145 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,15)(13,16)(17,22)(18,21)(19,23)(20,24);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,17)(13,18)(15,22)(16,21)(19,24)(20,23);(9,12,13,19)(10,15,16,23)(11,17,18,24)(14,20,21,22)Polycyclic presentation:
2^7|a^2=d,b^2=d,c^2=g,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 | #2030⋊C3, #11710966, #11711403, … |
| maximal subgroup of | #7427, #12204, #12206, … |
| maximal normal subgroup of | #7427, #12204, #12206, … |
| maximal quotient of | #10342, #10343, #10346, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 762);