#145734

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10564
commutator, Frattini, cosocle	C₂×#131	C₂³
prime radical	#3058	C₂²
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 308 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:

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

Polycyclic presentation:

2^9|a^2=i,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, 52930);