#143471

order	512
nilpotency class	4
isoclinism	isoclinic class of #143452
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

381 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 381 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(2,6)(3,9)(4,13)(8,14)(10,15)(11,12)(18,25)(20,37)(21,40)(22,23)(26,31)(27,41)(28,29)(30,38)(32,33)(34,35)(39,48)(42,44)(43,45)(46,47);(1,2,5,8)(3,10,12,15)(4,14,13,6)(7,11,16,9)(17,18)(19,31)(20,38)(21,41)(22,30)(23,29)(24,27)(25,37)(26,33)(28,40)(32,45)(34,43)(35,47)(36,42)(39,44)(46,48);(1,3)(2,7)(4,11)(5,12)(6,15)(8,16)(9,13)(10,14)(17,19,24,36)(18,26,27,44)(20,39,40,33)(21,32,37,48)(22,35,23,34)(25,42,41,31)(28,43,38,47)(29,46,30,45)

Polycyclic presentation:

2^9|b^2=gh,c^2=i,d^2=gi,e^2=gh,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eghi,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,e^c=egh,f^c=fgi,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, 50667);