#145943

order	512
nilpotency class	4
isoclinism	isoclinic class of #145943
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10593
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#472
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

201 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 201 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=de,b^2=dg,c^2=h,d^2=gi,e^2=hi,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,f^a=fg,g^a=gi,h^a=hi,c^b=cf,f^b=fi,d^c=dgi,e^c=eh,f^c=fgi,g^c=gi

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, 53139);