#6820948

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806551
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 ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1232 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,d^2=fi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei

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, 6728144);