#140584

order	512
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1955
prime radical	#532	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

307 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 90 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)(3,9)(4,12)(5,7)(6,10)(8,13)(11,15)(14,16)(17,18,21,27,23,28,39,52,22,26,38,41,40,50,67,69)(19,24,31,44,32,45,58,73,30,43,37,51,57,72,66,79)(20,25,35,47,36,49,54,76,34,48,60,68,63,77,80,70)(29,53,56,78,33,59,62,75,55,42,65,71,61,46,64,74);(1,3,4,10,5,11,13,16)(2,6,8,9,7,14,12,15)(17,19,21,37,23,32,39,66,22,30,38,31,40,57,67,58)(18,24,41,44,28,45,69,73,26,43,27,51,50,72,52,79)(20,33,60,62,36,55,80,65,34,61,35,64,63,29,54,56)(25,46,47,75,49,53,76,71,48,59,68,74,77,42,70,78);(2,7)(6,14)(8,12)(9,15)(17,20,22,34)(18,25,26,48)(19,29)(21,35,38,60)(23,36,40,63)(24,42)(27,47,41,68)(28,49,50,77)(30,55)(31,56)(32,33)(37,65)(39,54,67,80)(43,53)(44,71)(45,46)(51,78)(52,76,69,70)(57,61)(58,62)(59,72)(64,66)(73,74)(75,79)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=eg,d^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=di

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, 47780);