#148150

order	512
nilpotency class	4
isoclinism	isoclinic class of #13472
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2067
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)(5,6,9,17,11,18,32,14)(7,19,21,12,24,40,23,41)(8,25,27,49,29,44,30,51)(10,33,31,15,36,47,65,16)(13,35,45,28,46,63,43,64)(20,53,54,42,56,68,55,66)(22,38,57,48,59,39,34,50)(26,60,58,37,62,52,61,67);(1,2,3,4)(5,7,9,21,11,24,32,23)(6,12,14,19,18,41,17,40)(8,26,27,58,29,62,30,61)(10,34,31,22,36,57,65,59)(13,42,43,53,46,66,45,68)(15,48,33,38,16,50,47,39)(20,35,54,28,56,63,55,64)(25,37,51,60,44,67,49,52);(5,8,11,29)(6,13,18,46)(7,20,24,56)(9,30,32,27)(10,35,36,63)(12,37,41,67)(14,45,17,43)(15,49,16,51)(19,52,40,60)(21,55,23,54)(22,58,59,61)(25,33,44,47)(26,34,62,57)(28,31,64,65)(38,68,39,53)(42,50,66,48)

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=fhi,h^a=hi,c^b=cf,e^b=eh,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, 55346);