#6489593

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470019
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3012 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)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,18,21,24)(19,25,27,32)(20,28,30,23)(22,29,31,26);(1,3,6,13)(2,7,10,16)(4,12,5,11)(8,15,9,14)(17,19)(18,22)(20,29)(21,27)(23,32)(24,31)(25,28)(26,30);(1,4,6,5)(2,8,10,9)(3,11,13,12)(7,14,16,15)(20,30)(23,28)(25,32)(26,29);(1,5,6,4)(2,9,10,8)(3,12,13,11)(7,15,16,14);(2,10)(7,16)(8,9)(14,15)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6396789);