#6442851

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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×#3035	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2622 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)(11,16)(12,18)(13,28)(14,20)(15,19)(17,27)(21,31)(22,38)(23,33)(24,32)(25,37)(26,35)(29,36)(30,39)(34,40);(2,4)(9,11,14,23)(10,16,20,33)(12,25,27,40)(13,24,15,22)(17,34,18,37)(19,38,28,32)(21,29,39,26)(30,35,31,36);(5,6,7,8)(9,12)(10,17)(11,21)(13,26)(14,27)(15,29)(16,30)(18,20)(19,35)(22,25)(23,39)(24,40)(28,36)(31,33)(32,37)(34,38);(9,12)(10,18)(11,21)(13,26)(14,27)(15,29)(16,31)(17,20)(19,36)(22,25)(23,39)(24,40)(28,35)(30,33)(32,34)(37,38);(9,13,14,15)(10,19,20,28)(11,22,23,24)(12,26,27,29)(16,32,33,38)(17,35,18,36)(21,25,39,40)(30,37,31,34)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350047);