#6354222

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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1561 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18,22,26)(19,27,29,32)(20,25,21,24)(23,30,31,28);(1,3,5,13,6,11,4,12)(2,7,8,10,9,15,14,16);(1,2)(3,7)(4,14)(5,8)(6,9)(10,13)(11,15)(12,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,32)(26,31);(3,11)(7,15)(10,16)(12,13)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,27,31,32);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=i,e^2=h,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, 6261418);