#6349522

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349437
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	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2753 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:

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

Polycyclic presentation:

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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256718);