#143169

order	512
nilpotency class	4
isoclinism	isoclinic class of #143166
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10018
commutator, Frattini, cosocle	C₄×M₁₆	C₂³
prime radical	#575	D₄
LCS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

353 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 50 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=gi,e^2=gh,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,f^b=fi,g^b=gi,d^c=dh,e^c=egh,f^c=fi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50365);