#143157

order	512
nilpotency class	4
isoclinism	isoclinic class of #143154
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 ◁ #1746 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 408 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,g^b=gi,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, 50353);