#503820

order	512
nilpotency class	4
isoclinism	isoclinic class of #503819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#26107
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

406 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=e,c^2=ef,d^2=ei,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,g^c=ghi,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, 411016);