#511833

order	512
nilpotency class	4
isoclinism	isoclinic class of #511831
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32898
commutator, Frattini, cosocle	C₂×#41	C₂⁴
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2281 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)(3,17)(4,13)(5,25)(6,29)(7,15)(8,14)(9,16)(10,18)(11,24)(12,27)(19,48)(20,33)(21,45)(22,41)(23,37)(26,57)(28,43)(30,36)(31,38)(32,40)(34,60)(35,51)(39,54)(42,47)(44,49)(46,55)(50,59)(52,63)(53,56)(58,61)(62,64);(1,3,9,22)(2,10,16,36)(4,18,6,30)(5,19,28,50)(7,20,8,21)(11,33,38,45)(12,34,40,62)(13,41,29,17)(14,42,15,35)(23,46,26,58)(24,47,31,51)(25,55,43,61)(27,49,32,52)(37,59,57,48)(39,63,56,44)(53,64,54,60);(1,4)(2,11)(3,18)(5,23)(6,9)(7,31)(8,24)(10,33)(12,37)(13,15)(14,29)(16,38)(17,42)(19,46)(20,51)(21,47)(22,30)(25,56)(26,28)(27,54)(32,53)(34,59)(35,41)(36,45)(39,43)(40,57)(44,55)(48,62)(49,60)(50,58)(52,64)(61,63);(1,5)(2,12)(3,19)(4,23)(6,26)(7,27)(8,32)(9,28)(10,34)(11,37)(13,39)(14,25)(15,43)(16,40)(17,44)(18,46)(20,49)(21,52)(22,50)(24,53)(29,56)(30,58)(31,54)(33,59)(35,61)(36,62)(38,57)(41,63)(42,55)(45,48)(47,64)(51,60)

Polycyclic presentation:

2^9|b^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gi,h^a=hi,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,g^d=gi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419029);