#6353272

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353061
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 ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2932 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260468);