#6358718

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358586
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

1556 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265914);