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 | #33520 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #1752 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
799 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)(5,6,9,13,11,14,10,20)(7,8,15,18,16,19,12,17);(6,12,14,15)(7,13,16,20)(8,17)(9,10)(18,19);(2,4);(5,7)(6,8)(9,15)(10,12)(11,16)(13,18)(14,19)(17,20)Polycyclic presentation:
2^9|a^2=g,b^2=gh,e^2=h,g^2=i,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), #72241433, #72250638, … |
| maximal normal subgroup of | D4×AGL(1,ℤ/16), #72250638, #72250867, … |
| maximal quotient of | #72227144, #72227159, #72227162, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405316);