#94607

order	512
nilpotency class	6
isoclinism	isoclinic class of D₆₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₃₂
commutator	C₃₂	C₂×C₈
Frattini, cosocle	C₄×C₃₂	C₂²
socle	C₂²	#1415
prime radical	C₂×D₃₂	C₄
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

83 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=d,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, 1803);