#6351612

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351598
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	C₄²	C₂⁵
prime radical	#62823	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2696 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:

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

Polycyclic presentation:

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

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, 6258808);