#145215

order	512
nilpotency class	4
isoclinism	isoclinic class of #145214
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 268 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) = 256

A minimal faithful permutation representation — 3 generators on 80 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52411);