#143791

order	512
nilpotency class	4
isoclinism	isoclinic class of #143719
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
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 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 368 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=g,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fi,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, 50987);