#148614

order	512
nilpotency class	4
isoclinism	isoclinic class of #13504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2077
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11199
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

623 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,c^b=cf,f^b=fi,g^b=gi,e^c=ehi,f^c=fi,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, 55810);