#501269

order	512
nilpotency class	4
isoclinism	isoclinic class of #501262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61053	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(5,6,11,17,12,19,39,18)(7,20,23,46,9,30,32,48)(8,24,27,45,29,51,28,14)(10,33,37,16,38,49,58,50)(13,34,41,22,15,36,42,31)(21,53,55,67,25,62,56,52)(26,47,65,68,35,44,66,61)(40,54,60,63,43,57,59,64);(1,2)(3,4)(5,7,12,9)(6,13,19,15)(8,21,29,25)(10,34,38,36)(11,32,39,23)(14,40,45,43)(16,46,50,48)(17,42,18,41)(20,33,30,49)(22,37,31,58)(24,59,51,60)(26,63,35,64)(27,56,28,55)(44,53,47,62)(52,68,67,61)(54,66,57,65);(2,4)(5,8,12,29)(6,14,19,45)(7,21,9,25)(10,35,38,26)(11,28,39,27)(13,40,15,43)(16,47,50,44)(17,51,18,24)(20,52,30,67)(22,57,31,54)(23,56,32,55)(33,68,49,61)(34,64,36,63)(37,65,58,66)(41,59,42,60)(46,62,48,53);(5,9,12,7)(6,15,19,13)(8,25,29,21)(10,36,38,34)(11,23,39,32)(14,43,45,40)(16,48,50,46)(17,41,18,42)(20,49,30,33)(22,58,31,37)(24,60,51,59)(26,64,35,63)(27,55,28,56)(44,62,47,53)(52,61,67,68)(54,65,57,66)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408465);