#143171

order	512
nilpotency class	4
isoclinism	isoclinic class of #143170
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10026
commutator, Frattini, cosocle	#405	C₂³
prime radical	#33692	C₂
LCS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁

299 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 52 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50367);