Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) = #72216294 (order 1024) · Inn(G) = C22⋊C4 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22⋊C4 |
| commutator | C22 | C4×C8 |
| Frattini, cosocle | C22×C8 | C22 |
| socle | C22 | C22⋊C8 |
| prime radical | C23 | C2×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 20 points:
(5,6,7,11,8,12,14,18,9,10,13,16,15,17,19,20);(1,2,3,4)(6,10)(7,13)(12,17)(14,19)Polycyclic presentation:
2^7|a^2=d,b^2=e,d^2=f,f^2=g,b^a=bc,c^a=cg,d^b=dg44 conjugacy classes — view the full character table.
| maximal subgroup of | #7105, #7110, #7111, … |
| maximal normal subgroup of | #7105, #7110, #7111, … |
| maximal quotient of | #7139, #7143, #7146, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 109);