#148909

order	512
nilpotency class	4
isoclinism	isoclinic class of #140408
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1949
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11232
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

459 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=d,d^2=hi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,f^b=fi,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, 56105);