#148504

order	512
nilpotency class	4
isoclinism	isoclinic class of #13396
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2034
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

637 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:

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

Polycyclic presentation:

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

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, 55700);