#6704323

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#2416	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1256 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)(4,5)(7,8)(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,29,31)(27,33,35,28)(30,32,38,40)(34,36,39,37);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(26,31)(27,34)(28,36)(30,38)(33,37)(35,39);(2,5)(6,8)(25,27,29,35)(26,28,31,33)(30,39,38,34)(32,36,40,37);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,23)(22,24)(26,31)(27,35)(32,40)(34,39);(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,28)(26,27)(29,33)(30,36)(31,35)(32,39)(34,40)(37,38)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,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, 6611519);