#129392

order	512
nilpotency class	3
isoclinism	isoclinic class of #8465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1779
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

570 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 122 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,22,28,23,29,45,59,24,30,46,60,21,27,42,56)(19,31,35,62,33,51,67,57,36,49,43,72,34,25,47,50)(20,37,40,70,39,55,71,58,41,53,44,73,38,26,52,54)(32,63,65,77,64,76,74,80,66,48,69,75,68,61,78,79);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19)(18,25)(20,32)(21,34)(22,43)(23,33)(24,36)(26,48)(27,49)(28,57)(29,31)(30,51)(35,46)(37,61)(38,68)(39,64)(40,69)(41,66)(42,67)(44,65)(45,47)(50,60)(52,74)(53,76)(54,77)(55,63)(56,62)(58,75)(59,72)(70,80)(71,78)(73,79);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,23,39,24,41,21,38)(18,26,29,37,30,55,27,53)(19,32,33,64,36,66,34,68)(22,44,45,52,46,40,42,71)(25,48,31,61,51,63,49,76)(28,58,59,73,60,54,56,70)(35,69,67,78,43,65,47,74)(50,77,62,80,57,75,72,79)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gh,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=di,f^b=fgi,e^c=ei,f^c=fi

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, 36588);