#144539

order	512
nilpotency class	4
isoclinism	isoclinic class of #144538
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10351
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#454
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

264 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=dg,c^2=i,d^2=i,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,d^b=di,e^b=eh,f^b=fh,g^b=gi,h^b=hi,d^c=di,e^c=egi,f^c=fhi,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, 51735);