#147554

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	C₂×#575	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

993 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:

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

Polycyclic presentation:

2^9|a^2=g,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,d^b=di,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, 54750);