#6349170

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349137
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	#62816	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2862 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:

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

Polycyclic presentation:

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

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, 6256366);