#148814

order	512
nilpotency class	4
isoclinism	isoclinic class of #140094
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1919
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11220
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

471 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=dh,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=fhi,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, 56010);