#504338

order	512
nilpotency class	4
isoclinism	isoclinic class of #504336
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#26529
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	#577
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

426 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=efhi,b^2=fh,c^2=ef,d^2=ei,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,g^c=gh,h^c=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, 411534);