#94750

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	#1423
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

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

Polycyclic presentation:

2^9|a^2=d,b^2=cei,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

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, 1946);