#6349175

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	#62815	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2802 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)(3,10)(4,8)(5,14)(6,9)(7,11)(12,16)(13,15)(17,18,23,27)(19,28,31,32)(20,29,24,30)(21,26,22,25);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,16)(10,14)(17,19)(18,24)(20,27)(21,29)(22,30)(23,31)(25,28)(26,32);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,16,10)(17,20,23,24)(18,19,27,31)(21,32,22,28)(25,29,26,30);(17,21,23,22)(18,25,27,26)(19,29,31,30)(20,32,24,28);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,10,16,15)(17,21,23,22)(18,26,27,25)(19,30,31,29)(20,32,24,28)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=i,e^2=h,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, 6256371);