Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501156 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #24412 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,12)(8,17)(9,19,20,14)(10,15)(11,16,13,18);(1,2)(3,4)(5,7,10,8)(6,11,15,13)(9,16,20,18)(12,19,17,14);(2,4)(5,7,10,8)(6,12,15,17)(9,18,20,16)(11,19,13,14);(5,8,10,7)(6,13,15,11)(9,18,20,16)(12,14,17,19)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,f^b=fh,e^c=ehi,f^d=fh,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72312239, D4×#2198, #72312273, … |
| maximal normal subgroup of | #72312239, D4×#2198, #72312273, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408359);