#507706

order	512
nilpotency class	4
isoclinism	isoclinic class of #501524
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

457 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 457 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=g,d^2=ehi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=cg,f^b=fi,e^c=ei,f^c=fh,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, 414902);