#509176

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
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	#33527	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

759 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,d^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=eh,h^b=hi,f^c=fi,e^d=eh,f^d=fi,h^d=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, 416372);