#513278

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	C₂×D₁₆	C₂³
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

302 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,d^2=ei,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, 420474);