#503279

order	512
nilpotency class	4
isoclinism	isoclinic class of #500857
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1087 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6)(7,14)(8,23)(9,28)(10,17)(11,30)(12,18)(13,16)(15,19)(20,24)(21,26)(22,34)(25,32)(27,36)(29,35)(31,33);(5,6,13,28,11,17,29,20,12,18,24,35,10,30,9,16)(7,19,23,8,15,14,32,26,22,33,36,27,31,34,21,25);(1,2,3,4)(5,7)(6,14)(8,16)(9,21)(10,31)(11,15)(12,22)(13,23)(17,33)(18,34)(19,30)(20,27)(24,36)(25,35)(26,28)(29,32);(5,8,24,34,11,26,9,19,12,27,13,14,10,25,29,33)(6,15,35,21,17,22,16,23,18,31,28,32,30,7,20,36)

Polycyclic presentation:

2^9|b^2=eh,c^2=f,d^2=ehi,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410475);