#6681467

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1470 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=i,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6588663);