#6519814

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	#60799	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1938 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,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,f^b=fi,d^c=dh

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, 6427010);