#148640

order	512
nilpotency class	4
isoclinism	isoclinic class of #136628
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1875
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11200
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

621 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=dh,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fi,h^a=hi,c^b=cf,g^b=gi,d^c=di,e^c=eh,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, 55836);