#502104

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24266	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409300);