#6351649

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351632
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
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

2669 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271563, #72271570, #72271575
maximal normal subgroup of	#72271563, #72271570, #72271575

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258845);