#6435410

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432876
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2915 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:

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

Polycyclic presentation:

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

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, 6342606);