#6354958

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354913
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

1605 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)(3,5)(4,8)(6,11)(7,9)(10,12)(13,15)(14,16)(17,18,22,26)(19,23,28,32)(20,25,21,30)(24,29,31,27);(1,3,5,2,7,11,6,9)(4,10,12,8,13,16,14,15)(17,19)(18,23)(20,29)(21,27)(22,28)(24,30)(25,31)(26,32);(1,4)(2,8)(3,10)(5,12)(6,14)(7,13)(9,15)(11,16)(17,19)(18,24)(20,27)(21,29)(22,28)(23,30)(25,32)(26,31);(1,5,7,6)(2,3,9,11)(4,12,13,14)(8,10,15,16)(17,20,22,21)(18,25,26,30)(19,27,28,29)(23,24,32,31);(4,13)(8,15)(10,16)(12,14)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,d^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dhi,e^c=eh,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, 6262154);