#6442821

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2592 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350017);