#6354213

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354207
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
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

1602 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18,22,24)(19,26,25,21)(20,27,28,32)(23,29,31,30);(17,19,21,18,22,25,26,24)(20,27,29,23,28,32,30,31);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,20,22,28)(18,23,24,31)(19,27,25,32)(21,29,26,30);(2,7)(6,15)(8,12)(9,16)(18,24)(19,25)(23,31)(27,32);(20,28)(23,31)(27,32)(29,30)

Polycyclic presentation:

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

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, 6261409);