#6353496

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353473
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2688 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)(4,5)(8,14)(9,13)(11,12)(15,16)(17,18)(19,22)(20,28)(21,25)(23,27)(24,29)(26,31)(30,32);(1,2)(3,8)(4,13)(5,7)(6,9)(10,15)(11,16)(12,14)(17,19,22,18)(20,26,30,24)(21,27,23,25)(28,31,32,29);(1,2)(3,8)(4,7)(5,13)(6,9)(10,15)(11,14)(12,16)(17,20,22,30)(18,24,19,26)(21,29,23,31)(25,32,27,28);(1,3)(2,8)(4,11)(5,12)(6,10)(7,14)(9,15)(13,16)(17,21,22,23)(18,25,19,27)(20,29,30,31)(24,32,26,28);(3,10)(8,15)(11,12)(14,16)

Polycyclic presentation:

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

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, 6260692);