#500750

order	512
nilpotency class	4
isoclinism	isoclinic class of #500749
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24248
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61106	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1007 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=e,e^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=dh,e^c=ei,f^c=fh,g^c=ghi,h^c=hi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407946);