#6617615

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	#63010	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3025 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,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, 6524811);