#144165

order	512
nilpotency class	4
isoclinism	isoclinic class of #144131
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	C₂³	#496
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

378 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51361);