#491689

order	512
nilpotency class	4
isoclinism	isoclinic class of #491662
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 532 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:

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

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, 398885);