Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9303 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C24 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,4,3)(5,6)(7,9)(8,11)(10,12)(14,18)(15,16)(19,20);(6,9,11,10)(13,14)(15,20)(16,18)(17,19);(1,3,4,2)(13,14)(15,18)(16,20)(17,19)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,d^b=di,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72268657, #72268663 |
| maximal quotient of | #72268626 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 16938);