#148685

order	512
nilpotency class	3
isoclinism	isoclinic class of #2069
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#396
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11202
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

495 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=i,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,e^b=ei,f^b=fi,f^c=fi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55881);