#6347383

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347363
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	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

2846 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)(3,8)(4,9)(5,7)(6,10)(11,15)(12,16)(13,14)(17,18,22,26)(19,27,30,32)(20,25,21,24)(23,29,31,28);(2,7)(4,5)(8,14)(9,10)(12,13)(15,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,32)(26,31);(1,3)(2,8)(4,12)(5,13)(6,11)(7,14)(9,16)(10,15)(17,18)(19,23)(20,25)(21,24)(22,26)(27,29)(28,32)(30,31);(3,11)(8,15)(12,13)(14,16)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,27,31,32);(1,4,6,5)(2,9,10,7)(3,12,11,13)(8,16,15,14)(17,20,22,21)(18,25,26,24)(19,29,30,28)(23,27,31,32)

Polycyclic presentation:

2^9|a^2=i,d^2=i,e^2=h,f^2=h,b^a=bf,c^a=cg,d^a=di,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, 6254579);