#505146

order	512
nilpotency class	4
isoclinism	isoclinic class of #501868
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1771 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,6)(3,7,8,5)(9,10,14,16)(11,20,23,17)(12,18,27,31)(13,15,29,19)(21,32,37,36)(22,33,24,30)(25,38,39,34)(26,40,35,28);(1,3,4,8)(2,5,6,7)(9,11,10,17,14,23,16,20)(12,25,31,38,27,39,18,34)(13,22,19,33,29,24,15,30)(21,35,36,28,37,26,32,40);(9,12,14,27)(10,18,16,31)(11,21,23,37)(13,28,29,40)(15,26,19,35)(17,32,20,36)(22,38,24,34)(25,30,39,33);(2,6)(5,7)(9,13,16,15,14,29,10,19)(11,22,20,30,23,24,17,33)(12,26,18,28,27,35,31,40)(21,25,32,34,37,39,36,38)

Polycyclic presentation:

2^9|a^2=e,b^2=efh,c^2=i,d^2=fh,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=dh,g^c=ghi,h^c=hi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412342);