#6346644

order	512
nilpotency class	3
isoclinism	isoclinic class of #6346594
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	C₂²×C₄	C₂⁵
prime radical	#62781	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2956 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)(3,11,12,16)(4,9,5,8)(7,14,15,13)(23,30)(31,32);(1,3)(2,7)(4,13)(5,14)(6,12)(8,11)(9,16)(10,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,32)(29,30);(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,32)(26,30);(3,12)(7,15)(11,16)(13,14)(17,20,22,21)(18,24,26,25)(19,27,29,28)(23,31,30,32);(1,4,6,5)(2,8,10,9)(3,13,12,14)(7,11,15,16)(17,21,22,20)(18,25,26,24)(19,28,29,27)(23,32,30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284121, #72284146, #72285064
maximal normal subgroup of	#72284121, #72284146, #72285064

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253840);