#6548919

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1309 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6456115);