#6606278

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×#471
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3475	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1960 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6513474);