#6358763

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
prime radical	D₄×#41	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1627 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,e^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, 6265959);