#6694293

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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 ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1237 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)(26,30)(27,34)(31,33)(36,40);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,18)(21,24)(25,26,28,30)(27,35,34,32)(29,31,37,33)(36,39,40,38);(1,4,5,3)(2,6,7,8)(26,31)(27,34)(29,37)(30,33)(32,39)(35,38);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,27,28,34)(26,32,30,35)(29,36,37,40)(31,38,33,39);(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(26,33)(29,37)(30,31)(32,39)(35,38)(36,40)

Polycyclic presentation:

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

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, 6601489);