#6621385

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574100
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1819 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgi,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,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, 6528581);