#144526

order	512
nilpotency class	4
isoclinism	isoclinic class of #144526
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10349
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#453
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

276 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=d,c^2=g,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,h^a=hi,c^b=cf,e^b=eh,f^b=fh,g^b=gi,h^b=hi,d^c=di,e^c=egi,f^c=fh,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51722);