#497143

order	512
nilpotency class	4
isoclinism	isoclinic class of #483993
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

645 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=efh,d^2=ei,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=ce,f^b=fi,g^c=gi

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, 404339);