#499162

order	512
nilpotency class	4
isoclinism	isoclinic class of #498800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#469
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2054
prime radical	C₂×#3414	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|c^2=e,e^2=hi,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,e^b=eh,f^b=fhi,g^b=gh,h^b=hi,d^c=di,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406358);