#6556008

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1281 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=i,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,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, 6463204);