#6677473

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667748
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60572	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1652 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,6,9)(3,10,12,7)(4,13,5,8)(11,15,14,16)(17,18)(19,25)(20,34)(21,27)(22,39)(23,28)(24,29)(26,36)(30,48)(31,42)(32,44)(33,45)(35,41)(37,46)(38,47)(40,43);(1,3,6,12)(2,7,9,10)(4,11,5,14)(8,15,13,16)(17,19)(18,25)(20,35)(21,36)(22,32)(23,40)(24,33)(26,27)(28,43)(29,45)(30,37)(31,38)(34,41)(39,44)(42,47)(46,48);(17,20,24,31)(18,26,29,46)(19,30,33,21)(22,37,23,36)(25,41,45,47)(27,44,48,43)(28,42,39,34)(32,38,40,35);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,21,24,30)(18,27,29,48)(19,31,33,20)(22,38,23,35)(25,42,45,34)(26,44,46,43)(28,41,39,47)(32,37,40,36)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=f,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,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, 6584669);