#6617472

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573671
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62992	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3027 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6524668);