#6513585

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500474
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	#60805	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1934 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)(11,21,23,33)(12,26,28,36)(13,19,14,18)(16,25,32,24)(17,30,35,29)(22,34,27,31)(37,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,11,13,24,15,23,14,25)(10,16,18,21,20,32,19,33)(12,27,29,39,28,22,30,38)(17,34,26,40,35,31,36,37);(2,5)(3,7)(9,12)(10,17)(11,22)(13,29)(14,30)(15,28)(16,31)(18,26)(19,36)(20,35)(21,37)(23,27)(24,38)(25,39)(32,34)(33,40);(1,3,4,7)(2,6,5,8)(12,28)(17,35)(22,27)(26,36)(29,30)(31,34)(37,40)(38,39);(2,5)(6,8)(11,23)(16,32)(21,33)(22,27)(24,25)(31,34)(37,40)(38,39)

Polycyclic presentation:

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

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, 6420781);