Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) = C2×#511819 (order 1024) · Inn(G) = C22×D4 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×D4 |
| commutator | C4 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| socle | C2 | C2×C4×D4 |
| prime radical | #575 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
229 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,5,9,6,3,10,8)(4,7,13,15,12,11,16,14);(1,3)(2,6)(4,11)(5,8)(7,12)(9,10)(13,14)(15,16);(1,4,5,13,6,12,10,16)(2,7,9,15,3,11,8,14);(4,12)(7,11)(13,16)(14,15)Polycyclic presentation:
2^7|a^2=f,c^2=f,e^2=g,f^2=g,b^a=be,e^a=eg,e^b=eg,d^c=dg44 conjugacy classes — view the full character table.
| maximal subgroup of | #11669, #11670, #22898, … |
| maximal normal subgroup of | #11669, #11670, #22898, … |
| maximal quotient of | #17852, #17853, #17873, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1688);