#136804

order	512
nilpotency class	3
isoclinism	isoclinic class of #9431
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1869
prime radical	#532	C₂×C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

312 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 88 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 — 3 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=g,d^2=gi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=egi,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=ei,f^c=fg

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44000);