#151599

order	512
nilpotency class	5
isoclinism	isoclinic class of #151512
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2184
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	C₂×D₄	D₁₆
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

181 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 181 conjugacy classes of subgroups — too many to draw. Only its 48 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=fh,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,h^a=hi,d^b=df,e^b=eg,f^b=fh,h^b=hi,d^c=dg,e^c=eg

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58795);