#504060

order	512
nilpotency class	4
isoclinism	isoclinic class of #500736
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	#26302
prime radical	C₂×#135	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

423 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)(3,16)(4,22)(5,28)(6,14)(7,13)(8,31)(9,15)(10,19)(11,25)(12,29)(17,34)(18,51)(20,37)(21,43)(23,56)(24,59)(26,41)(27,44)(30,42)(32,36)(33,40)(35,53)(38,58)(39,61)(45,49)(46,52)(47,54)(48,62)(50,63)(55,60)(57,64);(1,3)(2,10,15,37)(4,23)(5,18)(6,19)(7,20)(8,32)(9,21)(11,38,41,56)(12,35,31,51)(13,16,14,43)(17,24)(22,54,44,64)(25,47)(26,57)(27,58)(28,46,42,36)(29,52)(30,53)(33,49)(34,55,63,40)(39,45,59,62)(48,60)(50,61);(1,4,6,25,9,27,7,26)(2,11,14,44,15,41,13,22)(3,17,19,48,21,50,20,49)(5,24,29,60,30,61,8,33)(10,34,43,62,37,63,16,45)(12,39,42,40,31,59,28,55)(18,47,52,58,53,57,32,23)(35,38,36,64,51,56,46,54);(1,5,6,29,9,30,7,8)(2,12,13,28,15,31,14,42)(3,18,19,52,21,53,20,32)(4,24,25,60,27,61,26,33)(10,35,16,46,37,51,43,36)(11,39,22,55,41,59,44,40)(17,47,48,58,50,57,49,23)(34,38,45,54,63,56,62,64)

Polycyclic presentation:

2^9|b^2=fh,c^2=efh,d^2=e,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,f^c=fi,g^c=gh,h^c=hi

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, 411256);