#145955

order	512
nilpotency class	4
isoclinism	isoclinic class of #145954
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10595
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#471
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

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

normal subgroups only

This group has 238 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:

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

Polycyclic presentation:

2^9|a^2=deh,b^2=d,c^2=i,d^2=g,e^2=h,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,d^c=dgi,e^c=eh,f^c=fg,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, 53151);