Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) = C2×#7622880 (order 1024) · Inn(G) = #117 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #117 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C2 | #381 |
| prime radical | C2×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ #117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
131 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,10,12,6)(4,9)(5,8)(7,11,14,16)(13,15);(1,3)(2,6)(4,11)(5,12)(7,9)(8,10)(13,16)(14,15);(1,3)(2,7,8,14)(4,11)(5,12)(6,9,10,15)(13,16)Polycyclic presentation:
2^7|a^2=f,c^2=d,e^2=g,b^a=bd,c^a=ce,d^a=dg,f^b=fg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #12719, #12720, #12721, … |
| maximal normal subgroup of | #12719, #12720, #12721, … |
| maximal quotient of | #8730, #8731, #8732, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 559);