#6490753

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2856	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1732 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)(4,7)(5,6)(9,10)(11,14)(12,15)(13,16)(17,18,20,24,21,23,29,31)(19,25,27,30,26,32,28,22);(2,6)(7,8)(9,11,14,10)(12,15,13,16)(17,19,21,26)(18,22,23,30)(20,28,29,27)(24,32,31,25);(1,3,5,4)(2,7,6,8)(10,11)(15,16)(19,26)(22,30)(25,32)(27,28);(9,12,14,13)(10,15,11,16)(19,26)(22,30)(25,32)(27,28);(1,4,5,3)(2,7,6,8)(9,13,14,12)(10,15,11,16)(18,23)(22,30)(24,31)(25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6397949);