#509166

order	512
nilpotency class	4
isoclinism	isoclinic class of #509163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33524	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

764 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=eh,g^b=gi,h^b=hi,g^c=gi,e^d=eh,h^d=hi

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, 416362);