#6581395

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	#24249	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1597 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)(17,18,22,23)(19,21,25,30)(20,24,27,31)(26,28,29,32);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(18,23)(19,21)(20,27)(25,30)(26,32)(28,29);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,19,22,25)(18,21,23,30)(20,28,27,32)(24,29,31,26);(1,4,5,3)(2,6,8,7)(17,20,22,27)(18,24,23,31)(19,26,25,29)(21,28,30,32);(17,21)(18,25)(19,23)(20,29)(22,30)(24,32)(26,27)(28,31)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=f,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,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, 6488591);