#6520039

order	512
nilpotency class	3
isoclinism	isoclinic class of #6502828
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

1684 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6427235);