#6520043

order	512
nilpotency class	3
isoclinism	isoclinic class of #6502937
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	#60920	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1668 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=hi,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^d=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, 6427239);