#504007

order	512
nilpotency class	4
isoclinism	isoclinic class of #500854
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33720	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

577 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)(9,10,17,22)(11,14,25,26)(12,23,19,15)(13,28,32,27)(16,20,33,21)(18,34,37,29)(24,39,40,31)(30,36,38,35);(2,5)(6,7)(9,11,15,21,16,19,14,22,17,25,23,20,33,12,26,10)(13,24,30,36,31,28,29,37,32,40,38,35,39,27,34,18);(9,12,14,21,33,25,15,10,17,19,26,20,16,11,23,22)(13,27,29,36,39,40,30,18,32,28,34,35,31,24,38,37);(1,3,4,8)(2,6,5,7)(9,13,23,38,16,31,26,34,17,32,15,30,33,39,14,29)(10,18,25,40,21,36,12,27,22,37,11,24,20,35,19,28)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411203);