#504789

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	#576
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

529 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2,4,5)(3,7,8,6)(9,10,14,22,16,24,28,58,18,26,15,23,17,25,43,46)(11,27,30,49,32,20,36,51,34,48,31,50,33,53,12,19)(13,37,38,60,45,59,44,56,41,54,39,55,40,57,42,21)(29,47,35,68,67,72,66,71,64,69,62,52,63,70,65,61);(2,5)(6,7)(9,11,18,34)(10,19,26,51)(12,15,36,14)(13,29,41,64)(16,33,17,32)(20,22,53,23)(21,47,56,69)(24,50,25,49)(27,58,48,46)(28,31,43,30)(35,44,62,42)(37,61,54,71)(38,65,39,66)(40,67,45,63)(52,57,68,59)(55,72,60,70);(9,12,18,36)(10,20,26,53)(11,28,34,43)(13,35,41,62)(14,33,15,32)(16,31,17,30)(19,46,51,58)(21,52,56,68)(22,49,23,50)(24,27,25,48)(29,39,64,38)(37,69,54,47)(40,66,45,65)(42,67,44,63)(55,61,60,71)(57,70,59,72);(1,3,4,8)(2,6,5,7)(9,13,18,41)(10,21,26,56)(11,29,34,64)(12,35,36,62)(14,42,15,44)(16,40,17,45)(19,47,51,69)(20,52,53,68)(22,57,23,59)(24,55,25,60)(27,61,48,71)(28,39,43,38)(30,65,31,66)(32,63,33,67)(37,58,54,46)(49,70,50,72)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=efh,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=hi,e^d=eh,f^d=fh,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411985);