#6803834

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1310 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fh,d^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6711030);