Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484039 |
| 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 | #33517 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #1752 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
847 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(6,10,12,20)(7,14)(8,18)(11,19,15,13)(16,17);(5,6,8,19,9,12,18,13)(7,15,14,10,17,11,16,20);(2,4);(6,11)(7,16)(8,18)(10,19)(12,15)(13,20)(14,17)Polycyclic presentation:
2^9|a^2=g,b^2=gh,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=de,e^b=eh,g^b=gi,h^b=hi,e^d=ehi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | D4×AGL(1,ℤ/16), #72241440, #72250443, … |
| maximal normal subgroup of | D4×AGL(1,ℤ/16), #72250443, #72250863, … |
| maximal quotient of | #72227144, #72227161, #72227163, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405312);