#500466

order	512
nilpotency class	4
isoclinism	isoclinic class of #499757
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

757 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=e,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,f^c=fi,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, 407662);