#6564067

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1597 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fgi,c^2=fg,d^2=fgi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6471263);