#147503

order	512
nilpotency class	4
isoclinism	isoclinic class of #147503
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	C₂⁴	C₂×C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,h^a=hi,d^b=dhi,e^b=eh,f^b=fh,h^b=hi,d^c=dh,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, 54699);