#145865

order	512
nilpotency class	4
isoclinism	isoclinic class of #145865
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10575
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#455
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

259 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 82 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)(4,9,19,35)(5,22)(6,10)(7,28)(11,39)(12,27)(13,45,46,62)(14,31)(15,49)(16,43)(17,33)(18,53,54,34)(20,56,55,41)(21,44,29,37)(23,58)(24,59,60,36)(25,42)(26,38)(30,47,61,52)(32,48)(40,51)(50,63,57,64);(1,3)(2,8)(4,18,19,54)(5,23)(6,17)(7,14)(9,34,35,53)(10,33)(11,40)(12,43)(13,20,46,55)(15,26)(16,27)(21,57,29,50)(22,58)(24,47,60,52)(25,48)(28,31)(30,36,61,59)(32,42)(37,64,44,63)(38,49)(39,51)(41,62,56,45);(1,4,7,29,6,19,27,21)(2,9,12,44,10,35,28,37)(3,13,16,52,17,46,14,47)(5,24,26,20,25,60,39,55)(8,30,31,45,33,61,43,62)(11,41,42,36,38,56,22,59)(15,50,23,18,51,57,48,54)(32,63,40,34,58,64,49,53)

Polycyclic presentation:

2^9|a^2=d,b^2=d,c^2=dh,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=eg,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, 53061);