#6349266

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	C₂³×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2920 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,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, 6256462);