#94608

order	512
nilpotency class	6
isoclinism	isoclinic class of D₆₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₃₂
commutator	C₃₂	C₂×C₈
Frattini, cosocle	C₄×C₃₂	C₂²
socle, prime radical	C₂²	#1415
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂ ◁ G
LECS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

77 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=i,c^2=eg,d^2=f,e^2=gh,g^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ce,e^b=eg,g^b=gh,h^b=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, 1804);