#6349568

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349529
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
prime radical	#62788	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2781 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)(13,15)(14,16)(17,18,22,24)(19,26,27,29)(20,28,23,31)(21,30,32,25);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19)(18,23)(20,24)(21,31)(22,27)(25,29)(26,30)(28,32);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,16,11)(17,20,22,23)(18,19,24,27)(21,26,32,29)(25,31,30,28);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,11,16,15);(2,10)(3,13)(8,9)(12,14)(18,24)(19,27)(25,30)(28,31)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=eh,f^a=fh,g^a=gi,e^b=eh,f^b=fh,g^b=gi,f^c=fi,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, 6256764);