#148165

order	512
nilpotency class	4
isoclinism	isoclinic class of #13417
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2041
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

674 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=i,e^2=h,f^2=hi,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=fhi,h^b=hi,d^c=di,e^c=ehi,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, 55361);