#6569503

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25401	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1416 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

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, 6476699);