#6590887

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
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	#2433	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1718 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6498083);