#6355264

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355203
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#3480	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1506 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262460);