#143323

order	512
nilpotency class	4
isoclinism	isoclinic class of #143322
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10054
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

444 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 80 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)(3,8)(4,17,20,53)(5,22)(6,27)(7,11)(9,19,36,56)(10,25)(12,29)(13,42,45,63)(14,47)(15,41)(16,32)(18,54,21,39)(23,58)(24,35,57,37)(26,52)(28,33)(30,44,34,62)(31,49)(38,50)(40,55)(43,51,46,61)(48,59,64,60);(1,3,7,16)(2,8,11,32)(4,13,20,45)(5,23,26,49)(6,28,29,15)(9,34,36,30)(10,38,40,47)(12,41,27,33)(14,25,50,55)(17,51,53,61)(18,46,21,43)(19,48,56,64)(22,58,52,31)(24,44,57,62)(35,59,37,60)(39,63,54,42);(1,4)(2,9)(3,13)(5,24)(6,18)(7,20)(8,30)(10,39)(11,36)(12,35)(14,48)(15,43)(16,45)(17,52)(19,55)(21,29)(22,53)(23,44)(25,56)(26,57)(27,37)(28,46)(31,61)(32,34)(33,59)(38,42)(40,54)(41,60)(47,63)(49,62)(50,64)(51,58)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50519);