#148788

order	512
nilpotency class	4
isoclinism	isoclinic class of #13396
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	#11220
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

469 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=dh,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=eh,f^b=fh,h^b=hi,d^c=di,e^c=ehi,f^c=fh,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, 55984);