#6580320

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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	#24450	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1683 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,25)(19,24,29,20)(21,26,23,27)(28,31,30,32);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19,22,29)(18,24,25,20)(21,30,23,28)(26,32,27,31);(10,14)(13,16)(18,25)(19,28)(20,31)(21,23)(24,32)(29,30);(1,4,5,3)(2,6,7,8)(17,20)(18,19)(21,31)(22,24)(23,32)(25,29)(26,30)(27,28);(17,21)(18,26)(22,23)(25,27)(28,30)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6487516);