#149376

order	512
nilpotency class	4
isoclinism	isoclinic class of #141878
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1988
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11373
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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=deh,d^2=h,e^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=dh,e^b=eh,g^b=gh,h^b=hi,d^c=dhi,e^c=ehi,g^c=gh,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, 56572);