#416940

order	512
nilpotency class	3
isoclinism	isoclinic class of #395581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1039 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)(3,7)(4,5)(6,8)(9,10,15,18,14,20,13,19)(11,21,24,31,25,34,23,16)(12,26,29,17,28,32,27,33)(22,30,37,39,38,40,36,35);(2,5)(6,7)(9,11,13,23,14,25,15,24)(10,16,18,21,20,31,19,34)(12,22,27,36,28,38,29,37)(17,30,32,39,33,40,26,35);(9,12,14,28)(10,17,20,33)(11,22,25,38)(13,29,15,27)(16,30,31,40)(18,26,19,32)(21,35,34,39)(23,37,24,36);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|a^2=fg,b^2=eh,c^2=i,d^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,g^b=gi,e^c=ei,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324136);