#513202

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
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

359 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)(9,10,15,21)(11,16,25,20)(12,26,30,60)(13,31,34,59)(14,35,37,40)(17,41,43,27)(18,44,47,33)(19,48,50,24)(22,52,29,64)(23,55,58,65)(28,62,63,71)(32,49,67,72)(36,45,54,70)(38,61,51,53)(39,68,46,57)(42,56,69,66);(1,3)(2,5)(4,8)(6,7)(9,11,15,25)(10,16,21,20)(12,22,30,29)(13,23,34,58)(14,24,37,48)(17,38,43,51)(18,39,47,46)(19,40,50,35)(26,52,60,64)(27,53,41,61)(28,54,63,36)(31,55,59,65)(32,56,67,66)(33,57,44,68)(42,49,69,72)(45,62,70,71);(9,12,15,30)(10,17,21,43)(11,22,25,29)(13,32,34,67)(14,36,37,54)(16,38,20,51)(18,45,47,70)(19,49,50,72)(23,56,58,66)(24,28,48,63)(26,59,60,31)(27,44,41,33)(35,42,40,69)(39,62,46,71)(52,65,64,55)(53,68,61,57);(3,8)(5,7)

Polycyclic presentation:

2^9|a^2=ei,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, 420398);