#148854

order	512
nilpotency class	4
isoclinism	isoclinic class of #13522
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2088
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11227
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

473 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=h,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,f^b=fi,e^c=eh,f^c=fi,g^c=ghi,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, 56050);