#6691017

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 ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1336 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)(3,6)(4,8)(5,7)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,30,31)(27,32,35,39)(28,37,36,33)(29,34,38,40);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,30,35)(26,32,31,39)(28,34,36,40)(29,37,38,33);(1,4,5,3)(2,6,7,8)(25,28)(26,33)(27,29)(30,36)(31,37)(32,40)(34,39)(35,38);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(27,35)(28,36)(32,39)(33,37);(11,19)(14,17)(18,24)(22,23)(25,29,30,38)(26,34,31,40)(27,36,35,28)(32,33,39,37)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=gi,d^2=f,e^2=i,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, 6598213);