#6348657

order	512
nilpotency class	3
isoclinism	isoclinic class of #6348613
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2856 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)(3,11)(4,8)(5,9)(6,10)(7,12)(13,16)(14,15)(17,18)(19,27)(20,25)(21,24)(22,26)(23,30)(28,32)(29,31);(1,3,6,14)(2,7,10,16)(4,12,5,13)(8,15,9,11)(17,19,22,29)(18,23,26,32)(20,28,21,30)(24,31,25,27);(1,2)(3,7)(4,9)(5,8)(6,10)(11,12)(13,15)(14,16)(18,24)(20,21)(23,31)(25,26)(27,32)(28,30);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(17,20,22,21)(18,25,26,24)(19,28,29,30)(23,27,32,31);(1,4,6,5)(2,9,10,8)(3,13,14,12)(7,15,16,11)(18,26)(19,29)(24,25)(28,30)

Polycyclic presentation:

2^9|b^2=h,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255853);