#143170

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
LCS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 305 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)(3,14)(4,21)(5,9)(6,30)(7,13)(8,16)(10,23)(11,28)(12,24)(15,47)(17,44)(18,43)(19,39)(20,46)(22,34)(25,40)(26,41)(27,56)(29,38)(31,51)(32,48)(33,53)(35,54)(36,58)(37,61)(42,60)(45,50)(49,64)(52,63)(55,59)(57,62);(1,3)(2,8)(4,22)(5,15)(6,19)(7,20)(9,32)(10,39)(11,31)(12,34)(13,35)(14,21)(16,23)(17,28)(18,24)(25,52)(26,51)(27,53)(29,50)(30,33)(36,44)(37,63)(38,62)(40,47)(41,45)(42,43)(46,56)(48,58)(49,61)(54,60)(55,64)(57,59);(2,9)(3,15)(4,23)(5,28)(6,24)(10,40)(11,12)(13,38)(14,43)(16,51)(17,18)(19,49)(20,50)(21,36)(22,57)(25,26)(27,60)(29,61)(30,37)(31,32)(33,34)(39,46)(41,42)(44,64)(48,53)(52,54)(55,56)(62,63)

Polycyclic presentation:

2^9|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, 50366);