#6354725

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354705
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	#60803	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1755 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)(3,10,13,15)(4,8,5,14)(7,11,16,12)(19,25)(20,27)(22,29)(24,31)(26,28)(30,32);(1,3,5,12,6,13,4,11)(2,7,8,10,9,16,14,15)(17,18)(19,22)(20,24)(21,23)(25,29)(26,30)(27,31)(28,32);(17,19,21,26)(18,22,23,30)(20,28,27,25)(24,32,31,29);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,16,15)(18,23)(22,30)(24,31)(29,32);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,15,16,10)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283862, #72283889
maximal normal subgroup of	#72283862, #72283889

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261921);