#137470

order	512
nilpotency class	4
isoclinism	isoclinic class of #137383
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#2184	C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

319 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 319 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=g,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=egh,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=egi,f^c=fh,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, 44666);