#6367494

order	512
nilpotency class	3
isoclinism	isoclinic class of #6367198
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1570 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:

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

Polycyclic presentation:

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

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, 6274690);