#6358835

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

1572 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,e^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, 6266031);