#148767

order	512
nilpotency class	4
isoclinism	isoclinic class of #140091
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1918
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11219
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

471 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)(3,17)(4,23,28,58)(5,15)(6,14)(7,31)(8,12)(9,16)(10,19)(11,26,43,33)(13,30)(18,50,53,40)(20,55)(21,45)(22,47)(24,59,56,36)(25,42,35,46)(27,41,61,48)(29,44)(32,37)(34,39)(38,57)(49,52,63,60)(51,62,54,64);(1,3,5,19,9,22,8,34)(2,10,12,17,16,39,15,47)(4,24,25,60,28,56,35,52)(6,21,7,32,29,57,30,20)(11,40,41,62,43,50,48,64)(13,45,44,55,31,38,14,37)(18,33,51,61,53,26,54,27)(23,49,46,59,58,63,42,36);(1,4,5,25,9,28,8,35)(2,11,12,41,16,43,15,48)(3,18,19,51,22,53,34,54)(6,27,7,33,29,61,30,26)(10,36,17,49,39,59,47,63)(13,46,44,58,31,42,14,23)(20,56,21,52,32,24,57,60)(37,40,45,62,55,50,38,64)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fh,h^b=hi,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, 55963);