#501931

order	512
nilpotency class	4
isoclinism	isoclinic class of #501930
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24965
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28249	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

485 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)(3,16)(4,22)(5,29)(6,14)(7,13)(8,15)(9,19)(10,26)(11,31)(12,30)(17,35)(18,45)(20,38)(21,47)(23,39)(24,40)(25,43)(27,46)(28,41)(32,44)(33,60)(34,59)(36,54)(37,53)(42,58)(48,51)(49,63)(50,61)(52,62)(55,56)(57,64);(1,3,6,19,8,21,30,20)(2,9,12,16,15,38,14,47)(4,23,26,49,28,57,25,56)(5,18,31,53,32,54,7,33)(10,39,22,55,43,64,41,63)(11,36,29,37,13,45,44,60)(17,34,51,27,52,58,50,24)(35,40,61,42,62,46,48,59);(1,4)(2,10)(3,17)(5,24)(6,25)(7,34)(8,28)(9,35)(11,40)(12,41)(13,46)(14,22)(15,43)(16,48)(18,49)(19,50)(20,51)(21,52)(23,53)(26,30)(27,32)(29,59)(31,58)(33,57)(36,39)(37,63)(38,62)(42,44)(45,64)(47,61)(54,56)(55,60);(1,5,30,7,8,32,6,31)(2,11,14,44,15,13,12,29)(3,18,20,33,21,54,19,53)(4,24,26,34,28,27,25,58)(9,36,47,60,38,45,16,37)(10,40,22,42,43,46,41,59)(17,49,51,57,52,56,50,23)(35,39,61,55,62,64,48,63)

Polycyclic presentation:

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

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, 409127);