#502195

order	512
nilpotency class	4
isoclinism	isoclinic class of #500730
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33527	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

731 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,e^2=hi,f^2=i,g^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=eh,h^b=hi,f^c=fi,g^c=gi,e^d=eh,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409391);