#143549

order	512
nilpotency class	4
isoclinism	isoclinic class of #143518
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#469
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

333 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 333 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=gi,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,f^a=fh,g^a=gi,c^b=cf,d^b=dg,f^b=fh,g^b=gi,d^c=dh,e^c=eh,f^c=fhi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50745);