#6354968

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354913
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	#60958	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1600 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)(3,10,11,16)(4,8,5,14)(7,12,15,13)(17,18,22,24)(19,21,26,31)(20,27,28,25)(23,29,32,30);(1,3,5,13,6,11,4,12)(2,7,8,10,9,15,14,16)(17,19,21,24,22,26,31,18)(20,25,29,32,28,27,30,23);(1,2)(3,7)(4,14)(5,8)(6,9)(10,13)(11,15)(12,16)(17,20)(18,23)(19,25)(21,29)(22,28)(24,32)(26,27)(30,31);(3,11)(7,15)(10,16)(12,13);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)(20,28)(23,32)(25,27)(29,30)

Polycyclic presentation:

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

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, 6262164);