Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | Aut(G) = #72237138 (order 1024) · Inn(G) = #459 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #459 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
85 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)(3,8)(4,6)(9,10)(11,16)(12,15)(13,14);(1,2,6,9,7,10,4,5)(3,8,14,16,12,15,11,13);(1,3)(2,8)(4,11)(5,13)(6,14)(7,12)(9,16)(10,15)Polycyclic presentation:
2^7|b^2=dg,d^2=g,e^2=fg,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ef,f^a=fg,e^c=ef,f^c=fg23 conjugacy classes — view the full character table.
| maximal subgroup of | #12310, #12311, #12314, … |
| maximal normal subgroup of | #12310, #12311, #12314, … |
| maximal quotient of | #12418, #12419, #12424, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 945);