Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D8 |
| commutator | C8 | C22×C4 |
| Frattini, cosocle | C2×C8 | C23 |
| prime radical | C22×D8 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4)(5,6)(7,12)(8,11)(9,18)(10,14)(13,16)(15,19)(17,20);(6,11)(7,9)(8,16)(12,13)(14,20)(15,17)(18,19);(5,7,10,9)(6,12,14,18)(8,15,17,16)(11,19,20,13)Polycyclic presentation:
2^7|a^2=e,c^2=g,d^2=fg,f^2=g,b^a=bd,d^a=df,f^a=fg,c^b=cg,d^b=df,f^b=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #12647, #12741, #12750, … |
| maximal normal subgroup of | #12647, #12741, #12750, … |
| maximal quotient of | #11593, #11649, #11661, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 872);