#6489612

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470037
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	#35534	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2831 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)(18,22)(20,27)(23,29)(24,28)(26,31)(30,32);(1,3,6,13)(2,7,10,16)(4,12,5,11)(8,15,9,14)(17,18)(19,23)(20,28)(21,24)(22,27)(25,30)(26,32)(29,31);(1,4,6,5)(2,8,10,9)(3,11,13,12)(7,14,16,15)(20,27)(22,28)(26,31)(29,32);(1,5,6,4)(2,9,10,8)(3,12,13,11)(7,15,16,14)(17,19,21,25)(18,23,24,30)(20,26,27,31)(22,29,28,32);(2,10)(7,16)(8,9)(14,15)(19,25)(23,30)(26,31)(29,32)

Polycyclic presentation:

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

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, 6396808);