#148162

order	512
nilpotency class	4
isoclinism	isoclinic class of #140211
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1936
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

666 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=i,e^2=h,f^2=h,g^2=i,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=ehi,f^b=fh,h^b=hi,d^c=di,e^c=ehi,f^c=fhi,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, 55358);