#143481

order	512
nilpotency class	4
isoclinism	isoclinic class of #143438
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	C₂³	#459
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

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

normal subgroups only

This group has 370 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) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50677);