#151123

order	512
nilpotency class	5
isoclinism	isoclinic class of #13694
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₁₆
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
socle, prime radical	C₂²	C₄×D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=ei,d^2=fh,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=cg,d^b=df,f^b=fh,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, 58319);