#501513

order	512
nilpotency class	4
isoclinism	isoclinic class of #501052
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33190	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

611 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=gi,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, 408709);