#6355564

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355552
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	#60965	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1566 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262760);