#504059

order	512
nilpotency class	4
isoclinism	isoclinic class of #504058
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#26300
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

428 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)(7,13)(8,12)(9,15)(10,19)(11,26)(17,49)(18,53)(20,39)(21,36)(22,37)(24,59)(25,61)(27,41)(28,43)(29,46)(31,45)(32,44)(33,48)(34,58)(35,52)(38,54)(40,60)(42,62)(47,51)(50,64)(55,63)(56,57);(1,3,7,20,9,22,6,19)(2,10,14,37,15,39,13,16)(4,24,26,56,29,60,28,50)(5,18,31,54,32,21,8,33)(11,40,23,57,43,59,46,64)(12,36,44,38,45,53,30,48)(17,34,51,27,52,62,55,25)(35,61,47,42,49,41,63,58);(1,4,6,26,9,29,7,28)(2,11,14,46,15,43,13,23)(3,17,20,55,22,52,19,51)(5,25,8,34,32,27,31,62)(10,35,16,47,39,49,37,63)(12,41,44,42,45,61,30,58)(18,50,54,60,21,56,33,24)(36,40,48,57,53,59,38,64);(1,5,7,31,9,32,6,8)(2,12,14,44,15,45,13,30)(3,18,20,54,22,21,19,33)(4,25,28,62,29,27,26,34)(10,36,37,38,39,53,16,48)(11,41,46,42,43,61,23,58)(17,50,55,60,52,56,51,24)(35,40,63,64,49,59,47,57)

Polycyclic presentation:

2^9|b^2=f,c^2=efh,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,c^b=cg,g^b=gh,h^b=hi,f^c=fi,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, 411255);