#6596835

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574100
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	#61070	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2088 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)(3,6)(4,7)(5,8)(9,10,15,20)(11,21,25,31)(12,26,28,33)(13,18,14,19)(16,23,32,24)(17,29,34,27)(22,36,39,30)(35,37,40,38);(1,3)(2,5)(4,8)(6,7)(9,11,14,23,15,25,13,24)(10,16,18,31,20,32,19,21)(12,22,29,37,28,39,27,38)(17,30,33,40,34,36,26,35);(3,8)(5,6)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,33)(19,26)(20,34)(21,35)(23,37)(24,38)(25,39)(31,40)(32,36);(1,3)(2,6)(4,8)(5,7)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,27,28,29)(16,21,32,31)(17,33,34,26)(22,37,39,38)(30,35,36,40);(3,8)(5,6)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,27,28,29)(16,21,32,31)(17,33,34,26)(22,37,39,38)(30,35,36,40)

Polycyclic presentation:

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

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, 6504031);