#6362143

order	512
nilpotency class	3
isoclinism	isoclinic class of #6362118
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

1656 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)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,18,22,25)(19,26,28,29)(20,23,30,27)(21,31,32,24);(1,3,5,11)(2,6,8,14)(4,10,12,16)(7,13,9,15)(17,19,22,28)(18,20,25,30)(21,23,32,27)(24,29,31,26);(2,7)(4,12)(6,13)(8,9)(10,16)(14,15)(17,20,22,30)(18,19,25,28)(21,26,32,29)(23,24,27,31);(17,20,22,30)(18,23,25,27)(19,24,28,31)(21,29,32,26);(2,8)(6,14)(7,9)(13,15)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6269339);