#500643

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	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

765 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=e,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, 407839);