#148311

order	512
nilpotency class	4
isoclinism	isoclinic class of #13402
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2034
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11095
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

649 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,f^b=fi,g^b=gi,d^c=di,e^c=ei,g^c=gi

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, 55507);