#6581316

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	#24244	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1636 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)(19,23)(20,22)(21,24)(25,32)(26,29)(27,30)(28,31);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(18,22)(19,25)(21,27)(23,31)(26,28)(29,32);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,19,20,26)(18,23,22,29)(21,28,27,25)(24,31,30,32);(1,4,5,3)(2,6,8,7)(19,25)(21,27)(23,32)(24,30)(26,28)(29,31);(17,19,20,26)(18,23,22,29)(21,28,27,25)(24,31,30,32)

Polycyclic presentation:

2^9|b^2=fgi,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, 6488512);