#6358799

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

1629 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)(7,15)(8,13)(9,16)(10,14)(17,18)(19,23)(20,27)(21,29)(22,25)(24,26)(28,31)(30,32);(1,2)(3,9)(4,13)(5,8)(6,12)(7,10)(11,15)(14,16)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,31,32,27);(17,20,22,28)(18,24,25,32)(19,26,21,30)(23,31,29,27);(1,3,5,11)(2,7,8,16)(4,14,12,10)(6,15,13,9)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,31,32,27);(2,8)(6,13)(7,16)(9,15)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=hi,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, 6265995);