#500739

order	512
nilpotency class	4
isoclinism	isoclinic class of #500738
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24236
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1021 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)(3,18)(4,24)(5,14)(6,16)(7,12)(8,13)(9,17)(10,21)(11,26)(15,30)(19,51)(20,37)(22,36)(23,48)(25,60)(27,47)(28,41)(29,43)(31,44)(32,49)(33,38)(34,45)(35,52)(39,53)(40,62)(42,61)(46,57)(50,54)(55,64)(56,63)(58,59);(1,3,6,21,9,23,8,33)(2,10,13,18,17,38,16,48)(4,25,27,58,28,54,34,52)(5,20,7,31,30,22,29,57)(11,39,40,51,42,64,49,63)(12,36,14,44,43,37,15,46)(19,32,53,61,56,62,55,26)(24,59,45,60,41,35,47,50);(1,4)(2,11)(3,19)(5,26)(6,27)(7,32)(8,34)(9,28)(10,35)(12,24)(13,40)(14,45)(15,47)(16,49)(17,42)(18,50)(20,52)(21,53)(22,58)(23,56)(25,31)(29,62)(30,61)(33,55)(36,51)(37,63)(38,59)(39,46)(41,43)(44,64)(48,60)(54,57);(1,5,8,29,9,30,6,7)(2,12,16,15,17,43,13,14)(3,20,33,57,23,22,21,31)(4,26,34,62,28,61,27,32)(10,36,48,46,38,37,18,44)(11,24,49,47,42,41,40,45)(19,52,55,54,56,58,53,25)(35,51,60,39,59,63,50,64)

Polycyclic presentation:

2^9|b^2=e,d^2=ei,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=gh,h^b=hi,f^c=fh,g^c=gh,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, 407935);