#6514392

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500489
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	#60919	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1698 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,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, 6421588);