Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | Aut(G) = C2×#72224760 (order 3072) · Inn(G) = #127 · Out(G) = C22×S4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #127 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| socle | C2 | #399 |
| prime radical | C24 | Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
111 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,4,8,5,7,10,14)(3,9,11,15,12,16,13,6);(1,3,10,11,5,12,4,13)(2,6,14,9,7,15,8,16);(2,7)(8,14)Polycyclic presentation:
2^7|a^2=d,b^2=de,d^2=g,b^a=bd,c^a=ce,f^a=fg,c^b=cf,d^b=dg,e^b=eg26 conjugacy classes — view the full character table.
| Fitting subgroup of | #2091⋊C3, #11710967, #11710968, … |
| maximal subgroup of | #7432, #7433, #12298, … |
| maximal normal subgroup of | #7432, #7433, #12298, … |
| maximal quotient of | #10659, #10660, #10661, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 823);