Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | Aut(G) = #6346637 (order 512) · Inn(G) = #447 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,6)(3,9)(4,12)(7,14)(8,15)(10,13)(11,16);(1,2)(3,10)(4,13)(5,7)(6,9)(8,12)(11,15)(14,16);(3,11)(6,14)(9,16)(10,15)Polycyclic presentation:
2^7|d^2=fg,f^2=g,b^a=bd,c^a=ce,d^a=df,f^a=fg,d^b=df,e^b=eg,f^b=fg,d^c=dg20 conjugacy classes — view the full character table.
| maximal subgroup of | #12114, #12116, #12117, … |
| maximal normal subgroup of | #12114, #12116, #12117, … |
| maximal quotient of | #12102, #12103, #12104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 922);