#500707

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1067 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=i,d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=di,f^c=fh,g^c=gh,h^c=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, 407903);