#491546

order	512
nilpotency class	4
isoclinism	isoclinic class of #491511
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂³	#493
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

489 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 489 conjugacy classes of subgroups — too many to draw. Only its 146 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 — 4 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fh,d^2=gi,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,c^b=cg,d^b=di

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, 398742);