#6357539

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357516
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60782	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1921 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264735);