#6359247

order	512
nilpotency class	3
isoclinism	isoclinic class of #6359226
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

1721 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,e^2=h,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,e^c=eh,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, 6266443);