#6355478

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355448
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	#60916	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1643 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=dhi,d^c=dh,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, 6262674);