#146187

order	512
nilpotency class	4
isoclinism	isoclinic class of #146186
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2092
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#478
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

197 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 197 conjugacy classes of subgroups — too many to draw. Only its 56 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=de,c^2=i,d^2=gi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,d^b=dg,e^b=egi,g^b=gi,h^b=hi,d^c=dh,e^c=eh,f^c=fi,g^c=gi,h^c=hi

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, 53383);