#6347391

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347363
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	#62834	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3016 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=h,e^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=di,f^a=fh,e^b=eh,f^b=fh,g^b=gi,d^c=dh,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, 6254587);