#6567782

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2011	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1402 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=fi,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=ei,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6474978);