#513226

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
prime radical	#33851	C₂
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

350 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)(11,16)(12,26)(13,31)(14,35)(15,21)(17,28)(18,33)(19,24)(20,25)(22,49)(23,52)(27,57)(29,63)(30,59)(32,47)(34,58)(36,44)(37,40)(38,50)(39,54)(41,60)(42,66)(43,62)(45,72)(46,61)(48,56)(51,71)(53,68)(55,65)(64,70)(67,69);(1,3)(2,5)(4,8)(6,7)(9,11,15,25)(10,16,21,20)(12,27,30,22)(13,23,34,55)(14,24,37,56)(17,41,43,38)(18,39,46,45)(19,40,48,35)(26,57,59,49)(28,60,62,50)(29,36,64,51)(31,52,58,65)(32,66,67,53)(33,54,61,72)(42,69,68,47)(44,70,71,63);(9,12,15,30)(10,17,21,43)(11,22,25,27)(13,32,34,67)(14,36,37,51)(16,38,20,41)(18,44,46,71)(19,47,48,69)(23,53,55,66)(24,29,56,64)(26,58,59,31)(28,61,62,33)(35,42,40,68)(39,63,45,70)(49,65,57,52)(50,72,60,54);(3,8)(5,7)(9,11,15,25)(10,16,21,20)(12,22,30,27)(13,23,34,55)(14,24,37,56)(17,38,43,41)(18,39,46,45)(19,40,48,35)(26,49,59,57)(28,50,62,60)(29,51,64,36)(31,52,58,65)(32,53,67,66)(33,54,61,72)(42,47,68,69)(44,63,71,70)

Polycyclic presentation:

2^9|a^2=e,b^2=i,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420422);