#6576436

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573635
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#576	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1893 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483632);