#6811131

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806815
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	#3061	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1721 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=h,d^2=f,e^2=fhi,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,g^c=gi

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, 6718327);