#513201

order	512
nilpotency class	5
isoclinism	isoclinic class of D₄×D₃₂
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

353 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)(17,27)(18,33)(19,24)(20,25)(22,49)(23,52)(28,62)(29,64)(30,58)(32,46)(34,57)(36,43)(37,40)(38,50)(39,54)(41,66)(42,61)(44,70)(45,59)(47,56)(48,60)(51,69)(53,68)(55,65)(63,71)(67,72);(1,3)(2,5)(4,8)(6,7)(9,11,15,25)(10,16,21,20)(12,22,30,29)(13,23,34,55)(14,24,37,56)(17,38,42,48)(18,39,45,44)(19,40,47,35)(26,49,58,64)(27,50,61,60)(28,51,63,36)(31,52,57,65)(32,53,67,66)(33,54,59,70)(41,46,68,72)(43,62,69,71);(9,12,15,30)(10,17,21,42)(11,22,25,29)(13,32,34,67)(14,36,37,51)(16,38,20,48)(18,43,45,69)(19,46,47,72)(23,53,55,66)(24,28,56,63)(26,57,58,31)(27,59,61,33)(35,41,40,68)(39,62,44,71)(49,65,64,52)(50,70,60,54);(3,8)(5,7)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420397);