#146103

order	512
nilpotency class	4
isoclinism	isoclinic class of #146102
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2073
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

197 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 197 conjugacy classes of subgroups — too many to draw. Only its 52 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:

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

Polycyclic presentation:

2^9|a^2=ehi,c^2=d,d^2=g,e^2=gi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fh,h^b=hi,e^c=eg,f^c=fgh,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53299);