Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) = #21483 (order 256) · Inn(G) = C22⋊C4 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C22⋊C4 |
| commutator | C22 | C2×C16 |
| Frattini, cosocle | C2×C16 | C22 |
| socle | C2 | C22⋊C16 |
| prime radical | C23 | C16 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
37 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3,8,4,9,12,20,5,10,13,21,14,22,25,30,6,7,11,17,15,18,23,27,16,19,24,28,26,29,31,32);(2,7)(3,11)(9,18)(10,19)(12,23)(13,24)(22,29)(25,31)Polycyclic presentation:
2^7|a^2=d,d^2=e,e^2=f,f^2=g,b^a=bc,c^a=cg,d^b=dg44 conjugacy classes — view the full character table.
| maximal subgroup of | #7336, #7337, #7364, … |
| maximal normal subgroup of | #7336, #7337, #7364, … |
| maximal quotient of | #7334, #7362, #7380, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 132);