#148653

order	512
nilpotency class	4
isoclinism	isoclinic class of #137665
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1895
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

629 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)(3,14,18,33)(4,20)(5,10)(6,28)(7,13)(8,16,34,19)(9,23)(11,26)(12,27)(15,44,47,53)(17,49,51,32)(21,56,45,61)(22,40)(24,54)(25,55)(29,37)(30,39)(31,57,62,50)(35,46,42,64)(36,58)(38,59)(41,60)(43,48,63,52);(1,3)(2,8)(4,21)(5,19)(6,17)(7,18)(9,35)(10,33)(11,32)(12,14)(13,34)(15,24)(16,27)(20,53)(22,47)(23,57)(25,45)(26,52)(28,43)(29,46)(30,51)(31,36)(37,62)(38,42)(39,63)(40,61)(41,49)(44,55)(48,60)(50,59)(54,56)(58,64);(1,4,7,25)(2,9)(3,15)(5,24,27,22)(6,29,30,58)(8,31,34,62)(10,37)(11,40)(12,36)(13,38)(14,42,33,35)(16,21)(17,50)(18,47)(19,45)(20,39)(23,26,59,60)(28,55)(32,44,49,53)(41,54)(43,56,63,61)(46,48)(51,57)(52,64)

Polycyclic presentation:

2^9|a^2=g,c^2=dh,d^2=i,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=di,f^b=fi,g^b=gi,d^c=di,e^c=ehi,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, 55849);