#147548

order	512
nilpotency class	3
isoclinism	isoclinic class of #2008
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#392
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10964
prime radical	#3449	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1005 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=di,g^b=gi,d^c=di,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, 54744);