#6514268

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	#60917	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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=hi,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, 6421464);