#141799

order	512
nilpotency class	4
isoclinism	isoclinic class of #141715
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1970
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#466
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

276 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 128

A minimal faithful permutation representation — 3 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=ehi,d^2=gi,e^2=hi,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg,d^b=di,f^b=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48995);