#148555

order	512
nilpotency class	4
isoclinism	isoclinic class of #140140
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1925
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11189
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

639 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:

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

Polycyclic presentation:

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

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, 55751);