#513222

order	512
nilpotency class	5
isoclinism	isoclinic class of #513173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

362 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=i,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,c^b=ci,d^b=de,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234341
maximal normal subgroup of	#72234341

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420418);