#145525

order	512
nilpotency class	4
isoclinism	isoclinic class of #145505
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#469
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

251 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 251 conjugacy classes of subgroups — too many to draw. Only its 60 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)(3,10)(4,12)(5,13)(6,9)(7,11)(8,15)(14,16)(17,18,24,30)(19,31,37,51)(20,38,43,75)(21,28,45,62)(22,27,47,60)(23,48,44,29)(25,34,54,33)(26,55,59,40)(32,65,69,73)(35,63,70,53)(36,72,49,52)(39,56,71,80)(41,58,46,74)(42,57,77,61)(50,76,79,68)(64,66,78,67);(1,3)(2,7)(4,8)(5,14)(6,12)(9,16)(10,13)(11,15)(17,19,24,37)(18,25,30,54)(20,39,43,71)(21,35,45,70)(22,33,47,34)(23,49,44,36)(26,56,59,80)(27,52,60,72)(28,31,62,51)(29,63,48,53)(32,46,69,41)(38,73,75,65)(40,76,55,68)(42,67,77,66)(50,61,79,57)(58,78,74,64);(1,4,5,7,6,10,15,16)(2,3,8,11,9,12,13,14)(17,20,22,46,23,42,45,55,24,43,47,41,44,77,21,40)(18,26,28,61,29,58,60,75,30,59,62,57,48,74,27,38)(19,32,35,71,36,68,34,67,37,69,70,39,49,76,33,66)(25,50,52,80,53,65,51,78,54,79,72,56,63,73,31,64)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=dh,d^2=gi,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=egi,f^b=fhi,g^b=gi,h^b=hi,e^c=eghi,f^c=fgh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52721);