#504061

order	512
nilpotency class	4
isoclinism	isoclinic class of #500737
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3325
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#26302
prime radical	C₂×#135	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

421 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)(4,22)(5,29)(6,14)(7,13)(8,32)(9,15)(10,19)(11,25)(12,30)(17,35)(18,53)(20,38)(21,45)(23,58)(24,43)(26,40)(27,42)(28,46)(31,44)(33,37)(34,41)(36,56)(39,59)(47,51)(48,55)(49,64)(50,62)(52,63)(54,57)(60,61);(1,3)(2,10,15,38)(4,23)(5,18)(6,19)(7,20)(8,33)(9,21)(11,39,42,58)(12,36,32,53)(13,16,14,45)(17,26)(22,57,46,64)(24,52)(25,54)(27,49)(28,59)(29,48,44,37)(30,55)(31,56)(34,51)(35,61,63,41)(40,62,43,47)(50,60);(1,4,6,25,9,28,7,27)(2,11,14,46,15,42,13,22)(3,17,19,50,21,52,20,51)(5,26,30,60,31,24,8,34)(10,35,45,62,38,63,16,47)(12,43,44,41,32,40,29,61)(18,54,55,59,56,49,33,23)(36,39,37,64,53,58,48,57);(1,5,6,30,9,31,7,8)(2,12,13,29,15,32,14,44)(3,18,19,55,21,56,20,33)(4,24,25,34,28,26,27,60)(10,36,16,48,38,53,45,37)(11,40,22,41,42,43,46,61)(17,49,50,23,52,54,51,59)(35,58,47,64,63,39,62,57)

Polycyclic presentation:

2^9|b^2=fh,c^2=efh,d^2=e,e^2=i,f^2=i,g^2=hi,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,d^c=di,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, 411257);