#147985

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:

(1,2,6,12,7,11,27,39,8,13,28,40,5,25,26,53)(3,14,19,41,18,31,47,56,17,32,46,57,16,9,29,33)(4,10,23,37,22,35,51,60,24,38,52,62,21,36,50,61)(15,42,44,63,45,30,54,55,43,34,58,59,20,48,49,64);(1,3,5,16,8,17,7,18)(2,9,11,14,13,31,25,32)(4,20,21,43,24,45,22,15)(6,19,26,29,28,46,27,47)(10,34,35,48,38,42,36,30)(12,33,39,41,40,56,53,57)(23,49,50,58,52,54,51,44)(37,59,60,64,62,63,61,55);(1,4)(2,10)(3,15)(5,21)(6,23)(7,22)(8,24)(9,30)(11,35)(12,37)(13,38)(14,34)(16,20)(17,43)(18,45)(19,44)(25,36)(26,50)(27,51)(28,52)(29,49)(31,48)(32,42)(33,55)(39,60)(40,62)(41,59)(46,58)(47,54)(53,61)(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,f^a=fi,c^b=cf,e^b=ei,f^b=fi,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, 55181);