#6349404

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349370
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	#62836	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2789 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)(3,8)(4,10)(5,7)(6,9)(11,16)(12,15)(13,14)(17,18,23,26)(19,27,30,31)(20,28,32,29)(21,24,22,25);(2,7)(4,5)(8,14)(9,10)(12,13)(15,16)(17,19)(18,20)(21,28)(22,29)(23,30)(24,27)(25,31)(26,32);(1,3)(2,8)(4,12)(5,13)(6,11)(7,14)(9,16)(10,15)(17,20)(18,19)(21,31)(22,27)(23,32)(24,29)(25,28)(26,30);(2,9)(3,11)(7,10)(12,13)(17,21,23,22)(18,24,26,25)(19,28,30,29)(20,27,32,31);(1,4,6,5)(2,10,9,7)(3,12,11,13)(8,15,16,14)(17,21,23,22)(18,24,26,25)(19,29,30,28)(20,31,32,27)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256600);