#6347559

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347544
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62822	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2950 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:

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

Polycyclic presentation:

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

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, 6254755);