#499392

order	512
nilpotency class	4
isoclinism	isoclinic class of #499029
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#482
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2077
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

665 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=e,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,d^b=di,e^b=ei,f^c=fh,g^c=gi,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, 406588);