#147989

order	512
nilpotency class	3
isoclinism	isoclinic class of #2008
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

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	#11042
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

667 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:

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

Polycyclic presentation:

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

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, 55185);