#146269

order	512
nilpotency class	4
isoclinism	isoclinic class of #146221
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2092
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#496
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

222 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=dgh,b^2=deg,c^2=g,d^2=gi,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,d^b=dg,e^b=eg,f^b=fh,h^b=hi,d^c=dg,e^c=eg,f^c=fh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53465);