#143655

order	512
nilpotency class	4
isoclinism	isoclinic class of #143654
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33780	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

378 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50851);