#143436

order	512
nilpotency class	4
isoclinism	isoclinic class of #143214
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1770 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

404 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 52 points:

(5,6,9,12)(7,13,15,20)(8,16,18,11)(10,17,19,14)(21,22)(23,36)(24,41)(25,30)(26,34)(27,33)(28,35)(29,38)(31,43)(32,44)(37,46)(39,51)(40,50)(42,47)(45,48)(49,52);(1,2)(3,4)(5,7,9,15)(6,10,12,19)(8,17,18,14)(11,20,16,13)(21,23,28,40)(22,29,35,49)(24,42,25,45)(26,37,27,39)(30,50,41,36)(31,51,32,46)(33,48,34,47)(38,44,52,43);(2,4)(22,30)(23,37)(24,43)(25,44)(26,27)(29,46)(31,33)(32,34)(35,41)(36,47)(38,52)(39,40)(48,50)(49,51)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=gi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ehi,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^c=eh,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50632);