#6350364

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350061
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 ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2678 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:

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

Polycyclic presentation:

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

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, 6257560);