#499037

order	512
nilpotency class	4
isoclinism	isoclinic class of #499029
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#482
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2034
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1047 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)(5,25)(6,12)(7,30)(8,13)(9,16)(10,23)(11,28)(15,44)(17,48)(18,43)(19,34)(21,56)(22,54)(24,55)(26,37)(27,40)(29,59)(31,39)(32,60)(33,50)(35,64)(36,57)(38,61)(41,46)(42,51)(45,62)(47,58)(49,63)(52,53);(1,3,7,19,8,16,6,18)(2,9,12,34,13,14,30,43)(4,21,24,45,22,52,31,15)(5,17,28,51,29,47,27,50)(10,35,38,32,36,63,37,41)(11,33,25,58,40,42,59,48)(20,53,39,62,54,56,55,44)(23,49,26,60,57,64,61,46);(1,4)(2,10)(3,15,16,45)(5,26)(6,24)(7,31)(8,22)(9,32,14,41)(11,39)(12,37)(13,36)(17,49,47,64)(18,21,19,52)(20,25)(23,28)(27,57)(29,61)(30,38)(33,56,42,53)(34,35,43,63)(40,55)(44,58,62,48)(46,50,60,51)(54,59);(4,22)(10,36)(15,45)(20,54)(21,52)(23,57)(24,31)(26,61)(32,41)(35,63)(37,38)(39,55)(44,62)(46,60)(49,64)(53,56)

Polycyclic presentation:

2^9|b^2=h,c^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,e^b=ei,d^c=di,f^c=fh,h^c=hi

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, 406233);