Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) = #72235124 (order 2048) · Inn(G) = C2×D4 · Out(G) = #3484 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
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 | C4×D4 |
| prime radical | C23 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,9,14,10,15,19,13)(7,11,8,17,16,12,18,20);(2,4)(5,7,9,8,10,16,19,18)(6,11,14,17,15,12,13,20);(5,8,9,16,10,18,19,7)(6,12,13,17,15,11,14,20)Polycyclic presentation:
2^7|a^2=f,b^2=f,c^2=e,e^2=g,f^2=g,b^a=bd,c^a=ce,e^a=eg,c^b=cg32 conjugacy classes — view the full character table.
| maximal subgroup of | #7125, #12944, #12956, … |
| maximal normal subgroup of | #7125, #12944, #12956, … |
| maximal quotient of | #9595, #9596, #9597, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 678);