#143925

order	512
nilpotency class	4
isoclinism	isoclinic class of #143904
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	#3169	C₂²
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

334 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 64 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)(3,10)(4,15)(5,8)(6,9)(7,12)(11,14)(13,16)(17,18,22,28)(19,30,33,44)(20,29,23,26)(21,25,37,42)(24,34,40,31)(27,35,43,36)(32,45,47,39)(38,46,41,48);(1,3,6,14)(2,7,9,16)(4,13,5,12)(8,11,15,10)(17,19,22,33)(18,24,28,40)(20,34,23,31)(21,32,37,47)(25,41,42,38)(26,44,29,30)(27,39,43,45)(35,48,36,46);(1,2)(3,11)(4,8)(5,15)(6,9)(7,12)(10,14)(13,16)(18,25)(20,23)(21,36)(24,38)(26,43)(27,29)(28,42)(30,45)(31,34)(32,46)(35,37)(39,44)(40,41)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=g,e^2=hi,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dg,f^b=fgi,d^c=di,e^c=eh,f^c=fgi,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, 51121);