#6358968

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358842
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

1600 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)(3,12)(4,11)(5,9)(6,8)(7,10)(13,16)(14,15)(17,18,22,25)(19,23,21,29)(20,27,28,31)(24,30,32,26);(1,3)(2,5)(4,9)(6,15)(7,13)(8,12)(10,14)(11,16)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,31,32,27);(1,4,7,6)(2,8,10,11)(3,9,13,15)(5,12,14,16)(17,20,22,28)(18,24,25,32)(19,26,21,30)(23,31,29,27);(1,5,7,14)(2,9,10,15)(3,8,13,11)(4,12,6,16)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,31,32,27);(2,10)(3,13)(8,11)(9,15)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,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, 6266164);