#150122

order	512
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

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	C₂²	C₂×#357
prime radical	C₂³	#357
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

201 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 201 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 80 points:

(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,30)(20,25)(21,39)(22,43)(23,29)(24,32)(26,41)(27,37)(28,38)(31,58)(33,64)(34,60)(35,59)(36,69)(40,55)(42,57)(44,52)(45,51)(46,61)(47,74)(48,67)(49,63)(50,71)(53,80)(54,62)(56,65)(66,78)(68,79)(70,77)(72,73)(75,76);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,23,34)(18,24,29,49)(20,35,38,31)(21,40,42,75)(22,44,45,68)(25,48,28,46)(26,52,54,79)(27,55,56,76)(30,57,60,39)(32,62,63,41)(33,65,66,37)(36,70,72,74)(43,47,51,77)(50,64,53,78)(58,69,59,73)(61,71,67,80);(3,9)(6,12)(10,15)(13,16)(17,20,23,38)(18,25,29,28)(19,31,34,35)(21,36,42,72)(22,37,45,65)(24,46,49,48)(26,50,54,53)(27,51,56,43)(30,58,60,59)(32,61,63,67)(33,44,66,68)(39,69,57,73)(40,74,75,70)(41,71,62,80)(47,76,77,55)(52,78,79,64)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=fh,e^2=g,f^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cgi,d^b=df,f^b=fh,h^b=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57318);