Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) = #72216294 (order 1024) · Inn(G) = C42 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C42 |
| commutator | C4 | C4×C8 |
| Frattini, cosocle | C22×C8 | C22 |
| socle | C22 | #97 |
| prime radical | C23 | M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2)(3,4)(5,6,8,13,9,11,17,18,10,14,16,20,7,12,15,19);(2,4)(6,11,14,12)(8,16)(13,19,20,18)(15,17)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=g,d^2=f,f^2=g,b^a=bc,e^a=eg,d^b=dg44 conjugacy classes — view the full character table.
| maximal subgroup of | #7109, #7116, #7117, … |
| maximal normal subgroup of | #7109, #7116, #7117, … |
| maximal quotient of | #7137, #7142, #7145, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 107);