#503984

order	512
nilpotency class	4
isoclinism	isoclinic class of #503983
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#26232
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

552 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)(3,16)(4,23)(5,30)(6,14)(7,11)(8,15)(9,19)(10,26)(12,31)(13,35)(17,36)(18,46)(20,40)(21,37)(22,38)(24,57)(25,47)(27,42)(28,44)(29,48)(32,45)(33,39)(34,43)(41,58)(49,52)(50,63)(51,60)(53,61)(54,62)(55,56)(59,64);(1,3,8,22)(2,9)(4,24)(5,18,32,21)(6,19,31,20)(7,33,35,54)(10,41,44,57)(11,37)(12,16)(13,46)(14,38)(15,40)(17,25)(23,56,48,63)(26,55)(27,53)(28,50)(29,58)(30,39)(34,51)(36,43,61,64)(42,49,47,60)(45,62)(52,59);(1,4,6,26,8,29,31,28)(2,10,12,23,15,44,14,48)(3,17,20,52,22,53,19,51)(5,25,7,34,32,27,35,59)(9,36,38,60,40,61,16,49)(11,42,30,43,13,47,45,64)(18,50,54,58,21,55,33,24)(37,57,62,56,46,41,39,63);(1,5,31,35,8,32,6,7)(2,11,14,45,15,13,12,30)(3,18,20,54,22,21,19,33)(4,25,28,59,29,27,26,34)(9,37,38,62,40,46,16,39)(10,42,48,64,44,47,23,43)(17,50,52,58,53,55,51,24)(36,57,60,56,61,41,49,63)

Polycyclic presentation:

2^9|b^2=fh,c^2=f,d^2=ei,e^2=i,f^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, 411180);