#148528

order	512
nilpotency class	4
isoclinism	isoclinic class of #13417
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2041
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

641 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)(4,20,24,43)(5,26)(6,31)(7,13)(8,16)(9,23,38,22)(10,29)(11,41)(12,28)(15,46,48,62)(17,36)(18,42)(19,34)(21,37,47,60)(25,58,55,52)(27,45)(30,44)(32,40)(33,53,61,49)(35,50)(39,56,51,57)(54,64,59,63);(1,3)(2,8)(4,21)(5,27)(6,32)(7,19)(9,37)(10,40)(11,42)(12,45)(13,36)(14,26)(15,23)(16,29)(17,41)(18,28)(20,33)(22,48)(24,47)(25,59)(30,50)(31,35)(34,44)(38,60)(39,64)(43,61)(46,58)(49,56)(51,63)(52,62)(53,57)(54,55);(1,4,5,22,28,55,41,57,7,24,30,23,6,25,29,56)(2,9,10,20,31,51,44,58,13,38,11,43,12,39,26,52)(3,15,16,47,32,53,50,54,19,48,17,21,18,49,27,59)(8,33,14,37,45,62,42,64,36,61,34,60,35,46,40,63)

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=hi,e^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fh,h^b=hi,e^c=ei,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, 55724);