#500645

order	512
nilpotency class	4
isoclinism	isoclinic class of #499736
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3155	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

777 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)(3,7)(4,5)(6,8)(9,10)(11,23)(12,29)(13,19)(14,21)(15,20)(16,39)(17,25)(18,32)(22,37)(24,57)(26,44)(27,43)(28,45)(30,67)(31,66)(33,54)(34,70)(35,51)(36,50)(38,56)(40,53)(41,48)(42,63)(46,64)(47,68)(49,69)(52,61)(55,59)(58,71)(60,72)(62,65);(2,5)(6,7)(9,11)(10,17)(12,30)(13,25)(14,27)(15,38)(16,40)(18,46)(19,23)(20,44)(21,52)(22,53)(24,33)(26,36)(28,37)(29,65)(31,62)(32,60)(34,63)(35,61)(39,45)(41,59)(42,48)(43,51)(47,72)(49,57)(50,56)(54,71)(55,70)(58,69)(64,68)(66,67);(9,12,13,31)(10,18,19,47)(11,24,25,58)(14,33,35,69)(15,32,36,68)(16,41,37,34)(17,42,23,55)(20,48,50,70)(21,29,51,66)(22,54,39,49)(26,62,38,30)(27,63,61,59)(28,64,40,60)(43,72,52,46)(44,57,56,71)(45,67,53,65);(1,3,4,8)(2,6,5,7)(10,19)(17,23)(18,47)(20,50)(21,51)(22,39)(29,66)(42,55)(43,52)(44,56)(45,53)(46,72)(48,70)(49,54)(57,71)(65,67)

Polycyclic presentation:

2^9|c^2=i,d^2=ei,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=ghi,h^b=hi,f^c=fh,g^c=ghi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407841);