Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) = #72236881 (order 1024) · Inn(G) = C22×D4 · Out(G) = C25 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×D4 |
| commutator | C4 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| socle | C2 | C2×C4×D4 |
| prime radical | C2×#135 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
226 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,10,5,19)(3,12,14,8,15,23,20,22)(4,9,18,24,16,25,17,29)(13,26,28,21,27,31,30,32);(1,3)(2,8)(4,13)(5,20)(6,14)(7,15)(9,21)(10,22)(11,23)(12,19)(16,27)(17,30)(18,28)(24,31)(25,32)(26,29);(1,4,6,18,7,16,5,17)(2,9,11,24,10,25,19,29)(3,13,14,28,15,27,20,30)(8,21,23,31,22,32,12,26);(2,10)(4,16)(8,22)(11,19)(12,23)(13,27)(17,18)(28,30)Polycyclic presentation:
2^7|a^2=f,c^2=f,e^2=g,f^2=g,b^a=be,d^a=dg,e^a=eg,e^b=eg,d^c=dg44 conjugacy classes — view the full character table.
| maximal subgroup of | #12829, #12830, #12942, … |
| maximal normal subgroup of | #12829, #12830, #12942, … |
| maximal quotient of | #17854, #17855, #17856, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1689);