#6596338

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573671
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	#63000	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2935 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:

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

Polycyclic presentation:

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

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, 6503534);