#6358738

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

1710 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)(19,22)(20,23)(21,25)(24,28)(26,29)(27,31)(30,32);(1,3)(2,7)(4,10)(5,11)(6,12)(8,16)(9,14)(13,15)(17,19,21,27)(18,22,25,31)(20,26,28,32)(23,29,24,30);(1,4,6,2)(3,10,12,7)(5,14,15,8)(9,13,16,11)(18,23)(20,28)(22,29)(24,25)(26,32)(30,31);(3,11)(5,15)(7,16)(8,14)(9,10)(12,13);(3,12)(7,10)(9,16)(11,13)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,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, 6265934);