#501029

order	512
nilpotency class	4
isoclinism	isoclinic class of #501028
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24354
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#62637	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

963 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408225);