#146105

order	512
nilpotency class	4
isoclinism	isoclinic class of #146104
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2073
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#488
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

189 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 189 conjugacy classes of subgroups — too many to draw. Only its 52 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=i,c^2=d,d^2=g,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fh,h^b=hi,e^c=egi,f^c=fgh,h^c=hi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53301);