#6685525

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682638
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60772	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592721);