#497089

order	512
nilpotency class	3
isoclinism	isoclinic class of #3192
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×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 428 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)(19,25,32,46,33,45,60,70,34,47,61,71,31,58,59,77)(20,35,38,63,39,49,65,72,36,62,64,78,37,26,48,50)(30,53,56,74,57,68,76,79,54,73,75,80,55,44,67,69);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,10)(14,15)(17,19,22,32,23,33,42,60,24,34,43,61,21,31,41,59)(18,25,28,46,27,45,51,70,29,47,52,71,40,58,66,77)(20,30,38,56,39,57,65,76,36,54,64,75,37,55,48,67)(26,44,50,69,35,53,63,74,49,68,72,79,62,73,78,80);(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)(19,30,31,55,34,54,33,57)(22,38,41,48,43,64,42,65)(25,44,45,53,47,68,58,73)(28,50,51,63,52,72,66,78)(32,56,59,67,61,75,60,76)(46,69,70,74,71,79,77,80);(1,5,6,4)(2,8,9,14)(3,11,12,13)(7,16,10,15)(20,36)(26,49)(30,54)(35,62)(37,39)(38,64)(44,68)(48,65)(50,72)(53,73)(55,57)(56,75)(63,78)(67,76)(69,79)(74,80)

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^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, 404285);