#6354469

order	512
nilpotency class	3
isoclinism	isoclinic class of #19384
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1747 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261665);