#6463247

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1568 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)(11,16)(12,14)(13,15)(17,18)(19,26)(20,24)(21,30)(22,25)(23,29)(27,32)(28,31);(1,3,4,8)(2,5,6,7)(11,12)(14,16)(19,27)(23,31)(26,32)(28,29);(9,11,13,12)(10,14,15,16)(17,19,21,29,22,27,20,28)(18,23,24,26,25,31,30,32);(9,12,13,11)(10,14,15,16)(17,20,22,21)(18,24,25,30)(19,28,27,29)(23,26,31,32);(17,20,22,21)(18,24,25,30)(19,28,27,29)(23,26,31,32)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi

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, 6370443);