#149009

order	512
nilpotency class	4
isoclinism	isoclinic class of #135850
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1854
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)(3,16,20,48,22,45,21,53)(4,23,28,56,29,41,27,55)(5,30,33,11,6,34,17,12)(9,35,37,19,38,52,46,31)(10,39,43,26,44,60,42,25)(18,40,50,36,51,62,24,57)(47,58,63,49,64,59,54,61);(1,3,17,35,32,20,6,19,8,22,33,52,7,21,5,31)(2,9,34,48,13,46,12,16,15,38,30,53,14,37,11,45)(4,24,39,49,28,51,26,58,29,50,60,61,27,18,25,59)(10,40,56,64,42,36,23,54,44,62,55,47,43,57,41,63);(1,4,5,25,7,27,33,60,8,29,6,26,32,28,17,39)(2,10,11,41,14,43,30,55,15,44,12,23,13,42,34,56)(3,18,19,49,21,50,35,59,22,51,31,61,20,24,52,58)(9,36,16,47,37,40,48,54,38,57,45,64,46,62,53,63)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=h,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,d^b=di,f^b=fi,g^b=gi,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, 56205);