#6355248

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	#3475	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1508 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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)(17,18,22,26)(19,23,28,32)(20,25,21,30)(24,29,31,27);(1,3,6,10,7,13,5,15)(2,8,11,12,9,4,14,16)(17,19)(18,23)(20,29)(21,27)(22,28)(24,30)(25,31)(26,32);(2,9)(3,13)(10,15)(11,14)(17,19)(18,24)(20,27)(21,29)(22,28)(23,30)(25,32)(26,31);(1,4)(2,10)(3,14)(5,12)(6,16)(7,8)(9,15)(11,13)(17,20,22,21)(18,25,26,30)(19,27,28,29)(23,24,32,31);(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,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, 6262444);