#6579789

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
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	#36434	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1751 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,25)(21,23)(22,26)(27,29)(28,30)(31,32);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(18,23)(19,27)(20,29)(22,31)(24,30)(25,28);(10,14)(13,16)(17,19,21,25)(18,24,23,20)(22,28,31,27)(26,30,32,29);(1,3,4,7)(2,6,5,8)(18,23)(19,28)(20,30)(22,31)(24,29)(25,27);(17,20,21,24)(18,25,23,19)(22,29,31,30)(26,27,32,28)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ehi,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6486985);