#6685515

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682658
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

1768 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)(4,5)(7,8)(9,10,14,18)(11,22,24,16)(12,27,30,34)(13,19,35,28)(15,21,29,20)(17,33,38,32)(23,36,40,37)(25,31,26,39);(2,5)(3,7)(9,11)(10,16)(12,28)(13,34)(14,24)(15,25)(17,36)(18,22)(19,30)(20,31)(21,39)(23,33)(26,29)(27,35)(32,40)(37,38);(9,12,29,33,14,30,15,32)(10,17,21,34,18,38,20,27)(11,23,26,19,24,40,25,28)(13,31,37,16,35,39,36,22);(2,5)(6,8)(10,18)(12,30)(16,22)(19,28)(20,21)(23,40)(31,39)(32,33);(1,3,4,7)(2,6,5,8)(9,13,14,35)(10,19,18,28)(11,17,24,38)(12,31,30,39)(15,36,29,37)(16,32,22,33)(20,23,21,40)(25,27,26,34)

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,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, 6592711);