#6358784

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈×AGL(1,ℤ/8)
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

1698 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265980);