#6554674

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535696
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61405	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fi,d^2=h,e^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,g^c=gi,e^d=eh

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, 6461870);