#149110

order	512
nilpotency class	3
isoclinism	isoclinic class of #2081
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#397
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11298
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=i,e^2=i,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=di,e^b=ei,g^b=gi,e^c=ei,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, 56306);