#6628062

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	#32913	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1772 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)(3,8)(4,6)(5,7)(9,10,13,14)(11,15,12,16)(18,23)(19,27)(20,22)(24,31)(28,29)(30,32);(1,3,5,4)(2,6,7,8)(10,14)(11,12)(17,18,21,23)(19,24,28,30)(20,25,22,26)(27,32,29,31);(9,11,13,12)(10,15,14,16)(17,19,21,28)(18,24,23,30)(20,27,22,29)(25,32,26,31);(1,4,5,3)(2,6,7,8)(11,12)(15,16)(17,20,21,22)(18,25,23,26)(19,27,28,29)(24,32,30,31);(9,12,13,11)(10,16,14,15)(18,23)(24,30)(25,26)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6535258);