#146189

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

Polycyclic presentation:

2^9|a^2=d,b^2=dei,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, 53385);