#6355445

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355407
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60916	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1602 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262641);