#504517

order	512
nilpotency class	4
isoclinism	isoclinic class of #500737
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3325
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#26657
prime radical	#13312	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

465 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=e,d^2=fh,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=di,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411713);