#491493

order	512
nilpotency class	4
isoclinism	isoclinic class of #491439
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

497 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 154 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)(7,12)(11,15)(13,16)(17,18,23,28)(19,24,30,43)(20,33,37,62)(21,38,40,66)(22,41,39,27)(25,46,48,35)(26,49,50,34)(29,51,53,73)(31,58,56,79)(32,59,57,45)(36,65,64,61)(42,67,68,54)(44,71,70,52)(47,63,60,72)(55,78,77,75)(69,76,74,80);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16)(17,19,23,30)(18,24,28,43)(20,29,37,53)(21,31,40,56)(22,32,39,57)(25,42,48,68)(26,44,50,70)(27,45,41,59)(33,51,62,73)(34,52,49,71)(35,54,46,67)(36,55,64,77)(38,58,66,79)(47,69,60,74)(61,75,65,78)(63,76,72,80);(17,20,34,63,22,36,40,46,23,37,49,72,39,64,21,35)(18,25,38,61,27,47,50,62,28,48,66,65,41,60,26,33)(19,29,52,76,32,55,56,67,30,53,71,80,57,77,31,54)(24,42,58,75,45,69,70,73,43,68,79,78,59,74,44,51);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(19,30)(24,43)(29,53)(31,56)(32,57)(42,68)(44,70)(45,59)(51,73)(52,71)(54,67)(55,77)(58,79)(69,74)(75,78)(76,80)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398689);