#6597521

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1781 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fhi,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308755, #72308792, #72308849
maximal normal subgroup of	#72308755, #72308792, #72308849

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6504717);