#6349305

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349298
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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2783 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:

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

Polycyclic presentation:

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

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, 6256501);