#6622271

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1445 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6529467);