#149055

order	512
nilpotency class	4
isoclinism	isoclinic class of #140203
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1934
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11292
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

445 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=hi,e^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=dh,f^b=fhi,g^b=ghi,h^b=hi,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, 56251);