#6499509

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469932
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1565 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,21)(14,17)(18,22)(19,23)(20,24)(25,26)(27,33)(28,36)(29,31)(30,35)(32,38)(34,40)(37,39);(9,11)(10,15)(12,20)(13,18)(14,19)(16,24)(17,23)(21,22)(25,27)(26,30)(28,37)(29,34)(31,39)(32,40)(33,36)(35,38);(2,5)(6,8)(9,12,14,13)(10,16,17,21)(11,18,19,20)(15,22,23,24)(27,34)(30,39)(33,40)(35,37);(1,3,4,7)(2,6,5,8)(11,19)(15,23)(18,20)(22,24)(27,34)(30,39)(33,40)(35,37);(9,13,14,12)(10,16,17,21)(11,18,19,20)(15,24,23,22)(26,31)(30,39)(32,36)(33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6406705);