#6597506

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1722 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fhi,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,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, 6504702);