#149005

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	#11268
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

423 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,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, 56201);