#6693679

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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

1326 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)(9,10,13,15)(11,14,12,16)(17,18)(19,23)(20,21)(22,24)(25,28)(26,29)(27,32)(30,31);(2,5)(3,7)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,29)(21,26)(22,27)(24,32)(25,30)(28,31);(9,12,13,11)(10,14,15,16)(17,20,21,18)(19,25,26,32)(22,24,31,30)(23,27,29,28);(2,5)(6,8)(18,20)(19,26)(24,30)(27,28);(1,3,4,7)(2,6,5,8)(19,27)(22,31)(23,32)(24,30)(25,29)(26,28)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,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, 6600875);