Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1355 |
| automorphism | Aut(G) = #152674 (order 512) · Inn(G) = #329 · Out(G) = D4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #329 |
| commutator | C42 | C2×C4 |
| Frattini, cosocle | #97 | C22 |
| prime radical | C22 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ G | |
| LECS | G ▷ #97 ▷ C2×C4 ▷ C2 ▷ C1 | |
39 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,9,5,10,18,28,6,11,17,29,20,8,25,26)(3,12,13,30,14,7,21,22,15,23,31,27,19,24,16,32);(1,3,6,15)(2,7,11,24)(4,16,17,21)(5,19,20,14)(8,23,10,12)(9,27,29,30)(13,18,31,25)(22,26,32,28)Polycyclic presentation:
2^7|a^2=d,b^2=g,c^2=f,d^2=e,e^2=g,b^a=bc,c^a=ce,f^a=fg,c^b=cf,d^b=defg,e^b=eg,d^c=dg20 conjugacy classes — view the full character table.
| maximal subgroup of | #7525, #7526, C2×#1356, … |
| maximal normal subgroup of | #7525, #7526, C2×#1356, … |
| maximal quotient of | #7117, #7130, #7266, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 88);