#6580020

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573477
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1584 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)(17,18,22,23)(19,24,29,32)(20,30,28,25)(21,26,31,27);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(18,23)(19,24)(21,26)(25,30)(27,31)(29,32);(10,14)(13,16)(17,19)(18,24)(20,31)(21,28)(22,29)(23,32)(25,26)(27,30);(1,3,4,7)(2,6,5,8)(17,20,22,28)(18,25,23,30)(19,27,29,26)(21,24,31,32);(17,21,22,31)(18,26,23,27)(19,28,29,20)(24,25,32,30)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,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, 6487216);