#6588951

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	#36471	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1607 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=fgi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,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, 6496147);