#143504

order	512
nilpotency class	4
isoclinism	isoclinic class of #143450
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	#3169	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

342 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,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=fgi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eghi,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

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50700);