#6357998

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357957
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	#62426	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1853 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)(3,10,8,5)(4,7,12,14)(11,16,15,13)(18,24)(19,26)(21,22)(27,28)(29,30)(31,32);(1,3,5,2,6,8,10,9)(4,11,13,7,12,15,16,14)(17,18,21,24,23,27,22,28)(19,25,29,32,30,20,26,31);(17,19,22,26,23,30,21,29)(18,25,28,31,27,20,24,32);(1,4,6,12)(2,7,9,14)(3,11,8,15)(5,13,10,16)(17,20,23,25)(18,26,27,29)(19,28,30,24)(21,31,22,32);(4,12)(7,14)(11,15)(13,16)(18,27)(20,25)(24,28)(31,32)

Polycyclic presentation:

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

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, 6265194);