Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) = #72237062 (order 4096) · Inn(G) = C2×D4 · Out(G) = #60399 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C22 | C2×C22⋊C4 |
| prime radical | #532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,9,12,10,13,8,18)(7,14,17,19,15,20,16,11);(5,7)(6,11,13,19)(8,16)(9,17)(10,15)(12,14,18,20);(1,2,3,4)(7,15)(11,19)(14,20)(16,17)Polycyclic presentation:
2^7|a^2=f,b^2=df,c^2=e,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,c^b=cg32 conjugacy classes — view the full character table.
| derived subgroup of | #72338409, #72338410, #72338411, … |
| Frattini subgroup of | #94534, #94535, #94536, … |
| maximal subgroup of | #7115, #7356, #7503, … |
| maximal normal subgroup of | #7115, #7356, #7503, … |
| maximal quotient of | #9640, #9641, #9642, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 695);