#6514264

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	#60559	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1744 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)(11,20)(12,24)(13,18)(14,30)(15,19)(16,22)(17,26)(21,37)(23,32)(25,36)(27,39)(28,35)(29,33)(31,40)(34,38);(1,2)(3,6)(4,5)(7,8)(9,11,14,22,15,23,13,29)(10,16,18,20,19,33,30,32)(12,25,28,40,26,21,27,38)(17,34,35,37,24,31,39,36);(2,5)(3,7)(9,12,14,28,15,26,13,27)(10,17,18,35,19,24,30,39)(11,21,22,38,23,25,29,40)(16,31,20,36,33,34,32,37);(1,3,4,7)(2,6,5,8)(12,26)(17,24)(21,25)(27,28)(31,34)(35,39)(36,37)(38,40);(2,5)(6,8)(9,13,15,14)(10,18,19,30)(11,22,23,29)(12,27,26,28)(16,32,33,20)(17,35,24,39)(21,38,25,40)(31,37,34,36)

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6421460);