Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484095 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#509 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(6,10,12,19)(7,14)(8,17)(11,18,20,13)(15,16);(5,6)(7,11)(8,18)(9,12)(10,15)(13,17)(14,19)(16,20);(1,2,3,4);(6,11)(7,15)(8,17)(10,18)(12,20)(13,19)(14,16)Polycyclic presentation:
2^9|a^2=g,c^2=f,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=de,e^b=ehi,g^b=gi,h^b=hi,e^d=ehi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72226987, #72227132, D4×AGL(1,ℤ/16), … |
| maximal normal subgroup of | #72226987, #72227132, D4×AGL(1,ℤ/16), … |
| maximal quotient of | #72226978, #72226987, #72226988, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405060);