#151591

order	512
nilpotency class	5
isoclinism	isoclinic class of #151520
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×D₁₆	D₄
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #2183 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

189 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 189 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58787);