#148619

order	512
nilpotency class	4
isoclinism	isoclinic class of #13497
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2075
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)(3,14,18,37)(4,20,24,55)(5,10)(6,28)(7,13)(8,16,33,19)(9,23,36,58)(11,26)(12,27)(15,44)(17,48,50,32)(21,56)(22,54,25,39)(29,41,59,35)(30,38)(31,61)(34,49)(40,60)(42,57)(43,47,64,51)(45,53)(46,63)(52,62);(1,3)(2,8)(4,21)(5,19)(6,17)(7,18)(9,34)(10,37)(11,32)(12,14)(13,33)(15,24)(16,27)(20,53)(22,45)(23,57)(25,52)(26,51)(28,43)(29,61)(30,50)(31,36)(35,42)(38,64)(39,56)(40,48)(41,63)(44,54)(46,58)(47,60)(49,59)(55,62);(1,4,5,22,7,24,27,25)(2,9,10,35,13,36,12,41)(3,15,16,45,18,21,19,52)(6,29,26,58,30,59,60,23)(8,31,14,42,33,34,37,63)(11,39,38,20,40,54,28,55)(17,49,47,46,50,61,51,57)(32,44,43,53,48,56,64,62)

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,e^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=di,g^b=gi,e^c=ehi,f^c=fi,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, 55815);