Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | Aut(G) = #6348988 (order 512) · Inn(G) = #476 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #476 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| prime radical | #135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
63 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,5)(3,11,13,8)(4,6)(9,10)(12,14,15,16);(1,2,4,5,7,10,6,9)(3,8,12,14,13,11,15,16);(1,3)(2,8)(4,12)(5,14)(6,15)(7,13)(9,16)(10,11)Polycyclic presentation:
2^7|a^2=df,b^2=d,d^2=g,e^2=f,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ef,f^a=fg,e^c=efg,f^c=fg23 conjugacy classes — view the full character table.
| maximal subgroup of | #12364, #12365, #12366, … |
| maximal normal subgroup of | #12364, #12365, #12366, … |
| maximal quotient of | #12823, #12950, #13116, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 962);