#509170

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	#25803	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)(5,6,13,20,36,21,37,38,11,7,10,16,12,17,41,34)(8,18,39,43,26,22,52,64,25,14,24,49,27,44,61,54)(9,28,50,46,32,51,35,15,31,29,30,45,33,47,40,19)(23,55,68,53,62,67,60,48,58,63,57,66,59,56,65,42);(5,7,11,6)(8,22,25,44)(9,29,31,28)(10,34,13,38)(12,17,36,21)(14,26,18,27)(15,30,19,50)(16,37,20,41)(23,56,58,67)(24,49,39,43)(32,51,33,47)(35,46,40,45)(42,60,48,65)(52,54,61,64)(53,68,66,57)(55,62,63,59);(2,4)(5,8)(6,14)(7,18)(9,23)(10,24)(11,25)(12,27)(13,39)(15,42)(16,43)(17,22)(19,48)(20,49)(21,44)(26,36)(28,63)(29,55)(30,57)(31,58)(32,62)(33,59)(34,64)(35,60)(37,52)(38,54)(40,65)(41,61)(45,53)(46,66)(47,67)(50,68)(51,56);(2,4)(5,9,11,31)(6,15,7,19)(8,23,25,58)(10,35,13,40)(12,32,36,33)(14,42,18,48)(16,47,20,51)(17,45,21,46)(22,53,44,66)(24,60,39,65)(26,59,27,62)(28,34,29,38)(30,41,50,37)(43,67,49,56)(52,57,61,68)(54,63,64,55)

Polycyclic presentation:

2^9|a^2=ei,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, 416366);