#6579521

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1347 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=f,d^2=i,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, 6486717);