#94749

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

Polycyclic presentation:

2^9|a^2=d,b^2=ce,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, 1945);