#6571608

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533075
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2858 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,f^a=fi,c^b=ch,f^b=fi,g^b=gi,f^c=fi,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, 6478804);