#94761

order	512
nilpotency class	6
isoclinism	isoclinic class of D₆₄
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₃₂
commutator	C₃₂	C₄²
Frattini, cosocle	C₂²×C₃₂	C₂²
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 2 generators on 72 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1957);