#6352836

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352701
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	#62905	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2858 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)(3,12)(4,9)(5,10)(6,15)(7,11)(8,13)(14,16)(17,18,22,25)(19,26,28,31)(20,24,21,29)(23,30,32,27);(1,3,7,4)(2,8,11,16)(5,13,6,14)(9,15,12,10)(17,19,22,28)(18,23,25,32)(20,27,21,30)(24,31,29,26);(17,18)(19,23)(20,24)(21,29)(22,25)(26,30)(27,31)(28,32);(1,4,7,3)(2,9,11,12)(5,13,6,14)(8,15,16,10)(17,18)(19,23)(20,29)(21,24)(22,25)(26,27)(28,32)(30,31);(1,5,7,6)(2,10,11,15)(3,13,4,14)(8,9,16,12)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260032);