#509157

order	512
nilpotency class	4
isoclinism	isoclinic class of #509155
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	#33522	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

779 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)(5,6,9,16,10,17,33,50,12,19,35,51,11,18,34,42)(7,20,23,39,24,55,62,41,26,56,57,40,25,49,31,13)(8,14,28,43,36,52,65,48,30,45,60,47,29,44,32,15)(21,53,58,67,63,68,61,66,59,54,22,38,27,46,64,37);(5,7,12,26)(6,13,19,41)(8,27,30,63)(9,31,35,62)(10,25,11,24)(14,38,45,67)(15,46,48,68)(16,49,51,55)(17,40,18,39)(20,50,56,42)(21,36,59,29)(22,60,58,28)(23,33,57,34)(32,64,65,61)(37,52,66,44)(43,54,47,53);(2,4)(5,8)(6,14)(7,21)(9,28)(10,36)(11,29)(12,30)(13,37)(15,42)(16,43)(17,52)(18,44)(19,45)(20,53)(22,57)(23,58)(24,63)(25,27)(26,59)(31,64)(32,34)(33,65)(35,60)(38,40)(39,67)(41,66)(46,49)(47,51)(48,50)(54,56)(55,68)(61,62);(2,4)(5,8)(6,15)(7,22)(9,32)(10,29)(11,36)(12,30)(13,38)(14,42)(16,44)(17,47)(18,43)(19,48)(20,54)(21,57)(23,59)(24,61)(25,64)(26,58)(27,31)(28,34)(33,60)(35,65)(37,40)(39,66)(41,67)(45,50)(46,49)(51,52)(53,56)(55,68)(62,63)

Polycyclic presentation:

2^9|a^2=e,b^2=i,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416353);