Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | Aut(G) = #72231699 (order 1024) · Inn(G) = #102 · Out(G) = C25 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #102 |
| commutator | C8 | C42 |
| Frattini, cosocle | C22×C8 | C22 |
| socle | C2 | #340 |
| prime radical | C23 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,11,8,13,7,24)(3,14,16,26,19,28,18,32)(4,20,22,29,23,12,6,10)(9,17,25,21,27,31,30,15);(1,3,6,17,7,18,23,15,8,19,22,31,5,16,4,21)(2,9,12,26,11,25,10,28,13,27,20,32,24,30,29,14)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=fg,d^2=g,e^2=f,f^2=g,b^a=bc,c^a=cf,e^a=ef,f^a=fg,c^b=cg32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1388, #12646, #12651, … |
| maximal normal subgroup of | C2×#1388, #12646, #12651, … |
| maximal quotient of | #7158, #7176, #7177, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 120);