#6366135

order	512
nilpotency class	3
isoclinism	isoclinic class of #6365694
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

1600 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6273331);