#143193

order	512
nilpotency class	4
isoclinism	isoclinic class of #143188
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	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 361 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)(6,11)(7,14)(10,16)(13,15)(17,18,21,28)(19,32,34,26)(20,25,22,40)(23,31,24,30)(27,39,29,37)(33,46,48,47)(35,43,36,45)(38,44,41,42);(6,15)(9,16)(18,25)(19,33)(20,37)(22,39)(23,24)(26,43)(27,30)(28,40)(29,31)(32,45)(34,48)(35,38)(36,41)(42,44);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(17,19,21,34)(18,26,28,32)(20,38,22,41)(23,36,24,35)(25,42,40,44)(27,46,29,47)(30,45,31,43)(33,37,48,39)

Polycyclic presentation:

2^9|a^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=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50389);