#6581314

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈×#120
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	#2537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1648 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)(11,13)(12,14)(15,16)(17,18)(19,24)(20,22)(21,25)(23,27)(26,31)(28,30)(29,32);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(18,22)(19,26)(21,28)(23,31)(24,32)(27,29);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,19,20,27)(18,23,22,24)(21,29,28,26)(25,31,30,32);(1,4,5,3)(2,6,8,7)(19,26)(21,28)(23,32)(24,31)(25,30)(27,29);(17,19,20,27)(18,24,22,23)(21,29,28,26)(25,32,30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6488510);