#6617619

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

2984 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=h,e^2=hi,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, 6524815);