Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) = #10586342 (order 512) · Inn(G) = C2×D4 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C2×D4 |
| commutator | C4 | C22×C8 |
| Frattini, cosocle | C2×C8 | C23 |
| socle | C2 | C2×C22⋊C8 |
| prime radical | C2×#41 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,11,6,12,20,29,7,9,21,25,4,10,18,27)(3,13,15,30,14,23,31,28,16,26,19,32,17,8,22,24);(1,3)(2,8)(4,17)(5,19)(6,14)(7,16)(9,23)(10,26)(11,28)(12,13)(15,21)(18,31)(20,22)(24,25)(27,30)(29,32);(2,9)(8,23)(10,12)(11,25)(13,26)(24,28)(27,29)(30,32)Polycyclic presentation:
2^7|a^2=e,d^2=g,e^2=f,f^2=g,b^a=bd,c^a=cg,d^b=dg,e^b=eg44 conjugacy classes — view the full character table.
| maximal subgroup of | #12609, #12618, #12685, … |
| maximal normal subgroup of | #12609, #12618, #12685, … |
| maximal quotient of | #11421, #11422, #11443, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 849);