#6349278

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349215
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
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2794 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)(3,10)(4,9)(5,14)(6,8)(7,12)(11,15)(13,16)(18,22)(19,25)(20,27)(24,28)(26,29)(30,32);(1,3,6,13)(2,7,8,15)(4,12,5,11)(9,16,14,10)(17,18,21,24)(19,23,26,31)(20,28,27,22)(25,30,29,32);(17,19,21,26)(18,23,24,31)(20,29,27,25)(22,30,28,32);(2,8)(7,15)(9,14)(10,16);(1,4,6,5)(2,9,8,14)(3,11,13,12)(7,10,15,16)(18,24)(22,28)(23,31)(30,32)

Polycyclic presentation:

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

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, 6256474);