Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | Aut(G) = #72231699 (order 1024) · Inn(G) = #106 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #106 |
| commutator | C8 | C22×C4 |
| Frattini, cosocle | C2×C8 | C23 |
| socle | C2 | C2×#106 |
| prime radical | C23 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12,15,26)(4,9)(5,18)(6,21)(7,11)(8,14,23,28)(10,19)(13,25,16,30)(17,24)(20,31)(22,27,32,29);(1,3,5,14,6,16,19,29,7,15,20,28,4,13,17,27)(2,8,10,12,9,22,18,25,11,23,24,26,21,32,31,30);(1,4,7,6)(2,9,11,21)(3,13,15,16)(5,17,20,19)(8,22,23,32)(10,18,24,31)(12,25,26,30)(14,27,28,29)Polycyclic presentation:
2^7|a^2=e,b^2=d,c^2=g,d^2=f,f^2=g,b^a=bd,d^a=df,f^a=fg,e^b=eg32 conjugacy classes — view the full character table.
| maximal subgroup of | #12668, #12678, #12832, … |
| maximal normal subgroup of | #12668, #12678, #12832, … |
| maximal quotient of | #11999, #12000, #12019, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 887);