#497091

order	512
nilpotency class	3
isoclinism	isoclinic class of #3198
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂³	C₂×#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

424 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 218 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 — 4 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ef,d^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=ce,d^b=di,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404287);