#6489607

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470028
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	#35190	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2889 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,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, 6396803);