#6353022

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352941
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

2744 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260218);