#148130

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	C₄	#2041
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11066
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

677 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)(4,23,28,46)(5,15)(6,14)(7,31)(8,12)(9,16)(10,19)(11,26,42,60)(13,30)(18,39,52,64)(20,54)(21,44)(22,47)(24,35,55,62)(25,45,34,41)(27,48,59,40)(29,43)(32,36)(33,38)(37,57)(49,51,58,56)(50,61,53,63);(1,3,5,19,9,22,8,33)(2,10,12,17,16,38,15,47)(4,24,25,51,28,55,34,56)(6,21,7,32,29,57,30,20)(11,39,40,63,42,64,48,61)(13,44,43,54,31,37,14,36)(18,26,50,59,52,60,53,27)(23,58,41,35,46,49,45,62);(1,4)(2,11)(3,18)(5,25)(6,27)(7,26)(8,34)(9,28)(10,35)(12,40)(13,45)(14,46)(15,48)(16,42)(17,49)(19,50)(20,55)(21,56)(22,52)(23,43)(24,32)(29,59)(30,60)(31,41)(33,53)(36,39)(37,61)(38,62)(44,63)(47,58)(51,57)(54,64)

Polycyclic presentation:

2^9|a^2=g,b^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=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, 55326);