#145917

order	512
nilpotency class	4
isoclinism	isoclinic class of #145917
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10589
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

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

normal subgroups only

This group has 198 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:

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

Polycyclic presentation:

2^9|a^2=de,b^2=d,c^2=h,d^2=g,e^2=hi,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,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, 53113);