#6355388

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355367
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	#60561	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1704 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)(7,11)(12,16)(13,15)(17,18)(19,28)(20,24)(21,26)(22,27)(23,31)(25,30)(29,32);(1,3,5,11,6,13,4,12)(2,7,8,10,9,16,14,15)(17,19,22,20)(18,24,27,28)(21,30,23,29)(25,26,32,31);(17,20,22,19)(18,25,27,32)(21,30,23,29)(24,26,28,31);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,16,10)(17,21,22,23)(18,26,27,31)(19,29,20,30)(24,25,28,32);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,10,16,15)(18,27)(24,28)(25,32)(26,31)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=hi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=dhi,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, 6262584);