#6499859

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499855
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#35346
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3755 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261107, #72261154, C₂×#6499859
maximal normal subgroup of	#72261107, #72261154, C₂×#6499859
maximal quotient of	#72267103, #72306508, #72306512, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6407055);