#148784

order	512
nilpotency class	4
isoclinism	isoclinic class of #13460
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2058
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11220
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

473 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=dh,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=eh,f^b=fh,h^b=hi,d^c=di,e^c=eh,f^c=fh,g^c=gi,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, 55980);