#147514

order	512
nilpotency class	4
isoclinism	isoclinic class of #147512
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10960
commutator	C₂²×C₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

803 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efi,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gh,h^a=hi,d^b=dhi,e^b=ehi,f^b=fhi,h^b=hi,d^c=dhi,e^c=eh,f^c=fh,h^c=hi,f^d=fi,g^d=gi,f^e=fi,g^e=gi,g^f=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54710);