#148810

order	512
nilpotency class	4
isoclinism	isoclinic class of #13404
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	#11220
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

467 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)(3,16,18,9,22,41,21,42)(4,23,27,48,28,59,25,50)(6,32,29,12,7,35,30,13)(10,37,44,61,45,34,46,26)(17,52,54,43,55,63,53,64)(19,39,33,49,20,40,36,47)(24,51,56,38,60,62,57,58);(1,3,5,18,8,22,31,21)(2,9,11,16,15,42,14,41)(4,24,25,57,28,60,27,56)(6,33,29,20,7,36,30,19)(10,43,44,63,45,64,46,52)(12,47,32,40,13,49,35,39)(17,37,53,26,55,34,54,61)(23,58,48,51,59,38,50,62);(1,4,8,28)(2,10,15,45)(3,17)(5,27,31,25)(6,34,7,37)(9,38)(11,46,14,44)(12,48,13,50)(16,51)(18,54)(19,56)(20,57)(21,53)(22,55)(23,35,59,32)(24,36)(26,30,61,29)(33,60)(39,52)(40,63)(41,62)(42,58)(43,49)(47,64)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=dh,d^2=i,e^2=hi,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=eh,f^c=fhi,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, 56006);