#6576440

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×#578	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1847 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)(9,10,15,22)(11,12,27,18)(13,20,32,38)(14,21,34,25)(16,24,17,23)(19,30,35,29)(26,28,39,33)(31,37,36,40);(2,5)(6,8)(9,11,15,27)(10,18,22,12)(13,25,32,21)(14,20,34,38)(16,19,17,35)(23,30,24,29)(26,31,39,36)(28,40,33,37);(9,12,15,18)(10,19,22,35)(11,24,27,23)(13,31,32,36)(14,28,34,33)(16,30,17,29)(20,25,38,21)(26,37,39,40);(1,3,4,7)(2,6,5,8)(9,13)(10,20)(11,25)(12,14)(15,32)(16,33)(17,28)(18,34)(19,37)(21,27)(22,38)(23,39)(24,26)(29,31)(30,36)(35,40);(9,14)(10,21)(11,26)(12,28)(13,29)(15,34)(16,31)(17,36)(18,33)(19,20)(22,25)(23,40)(24,37)(27,39)(30,32)(35,38)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^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, 6483636);