#145733

order	512
nilpotency class	4
isoclinism	isoclinic class of #145733
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10564
commutator, Frattini, cosocle	C₂×#131	C₂³
prime radical	#33765	C₂
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

316 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 58 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)(3,16)(4,9)(5,28)(6,12)(7,31)(8,15)(10,19)(11,23)(13,30)(14,24)(17,40)(18,52)(20,47)(21,39)(22,48)(25,43)(26,33)(27,36)(29,44)(32,53)(34,59)(35,57)(37,55)(38,63)(41,51)(42,50)(45,49)(46,54)(56,62)(58,60)(61,64);(2,9)(3,17)(4,23)(5,29)(6,30)(7,24)(10,37)(11,14)(12,34)(13,33)(15,36)(16,45)(18,53)(19,21)(20,50)(22,51)(25,59)(27,57)(28,60)(31,35)(32,39)(38,48)(40,52)(41,62)(42,46)(43,44)(47,61)(49,54)(55,63)(56,64);(1,3,4,18,24,21,57,55,8,22,27,56,7,20,23,54)(2,10,11,38,31,41,36,61,15,42,35,45,14,40,9,32)(5,19,25,53,6,17,26,49,30,50,59,64,29,51,58,63)(12,16,43,46,28,47,60,62,44,48,34,37,13,39,33,52)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52929);