#6583014

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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	#24782	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1437 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)(19,26)(20,29)(23,30)(27,31);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,18,21,25)(19,27,26,31)(20,30,29,23)(22,28,32,24);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,19,21,26)(18,20,25,29)(22,30,32,23)(24,31,28,27);(1,3,5,4)(2,7,8,6)(17,20,21,29)(18,23,25,30)(19,28,26,24)(22,31,32,27);(1,4,5,3)(2,6,8,7)(18,24)(20,31)(22,32)(23,30)(25,28)(27,29)

Polycyclic presentation:

2^9|b^2=fg,c^2=fi,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490210);