#499596

order	512
nilpotency class	4
isoclinism	isoclinic class of #499029
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#482
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23758
prime radical	#3067	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

436 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)(3,10)(4,20)(5,12)(6,13)(7,29)(8,30)(9,14)(11,23)(15,28)(16,48)(17,35)(18,38)(19,39)(21,58)(22,47)(24,43)(25,45)(26,57)(27,44)(31,61)(32,41)(33,36)(34,52)(37,54)(40,53)(42,49)(46,59)(50,64)(51,63)(55,62)(56,60);(1,3,7,18,6,19,9,33)(2,10,14,36,13,39,29,38)(4,21,23,50,25,60,24,53)(5,17,8,31,27,54,28,59)(11,40,20,56,43,64,45,58)(12,35,15,46,44,37,30,61)(16,49,52,22,51,32,55,26)(34,47,48,42,62,57,63,41);(1,4)(2,11)(3,16)(5,26)(6,25)(7,24)(8,32)(9,23)(10,34)(12,42)(13,43)(14,45)(15,47)(17,53)(18,55)(19,51)(20,29)(21,59)(22,27)(28,49)(30,57)(31,60)(33,52)(35,58)(36,63)(37,56)(38,48)(39,62)(40,61)(41,44)(46,64)(50,54);(1,5,7,8,6,27,9,28)(2,12,29,30,13,44,14,15)(3,17,18,31,19,54,33,59)(4,22,24,49,25,26,23,32)(10,35,38,61,39,37,36,46)(11,41,20,47,43,42,45,57)(16,50,55,21,51,53,52,60)(34,56,48,64,62,58,63,40)

Polycyclic presentation:

2^9|b^2=f,d^2=eh,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gh,h^b=hi,d^c=di,e^c=ei,f^c=fi,g^c=ghi,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, 406792);